%------------------------------------------------------------------------------
% File     : ITP277^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00371_024092
% 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_00371_024092 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11481 (4686 unt;1316 typ;   0 def)
%            Number of atoms       : 32221 (11947 equ;   0 cnn)
%            Maximal formula atoms :   71 (   3 avg)
%            Number of connectives : 118110 (3292   ~; 543   |;2555   &;98472   @)
%                                         (   0 <=>;13248  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  128 ( 127 usr)
%            Number of type conns  : 4875 (4875   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1192 (1189 usr; 104 con; 0-8 aty)
%            Number of variables   : 27245 (2185   ^;24068   !; 992   ?;27245   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 16:12:26.953
%------------------------------------------------------------------------------
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thf(ty_n_t__VEBT____Definitions__OVEBT,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

thf(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

thf(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

thf(ty_n_t__Option__Ooption_I_Eo_J,type,
    option_o: $tType ).

thf(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

thf(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

thf(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

thf(ty_n_t__String__Oliteral,type,
    literal: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_t__Rat__Orat,type,
    rat: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (1189)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

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

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

thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: rat > rat ).

thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: real > real ).

thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: real > int ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

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_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_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__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_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__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_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__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__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_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__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_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__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

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

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

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

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

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

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

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

thf(sy_c_Bit__Operations_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__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__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
    comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

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

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

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
    comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: set_int > int ).

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__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

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,
    csqrt: complex > complex ).

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( real > 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_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__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

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

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

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

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

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

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

thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
    euclid3395696857347342551nt_int: int > int ).

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

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

thf(sy_c_Extended__Nat_Oenat_OAbs__enat,type,
    extended_Abs_enat: option_nat > extended_enat ).

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

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
    extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nat__Oenat,type,
    extend5688581933313929465d_enat: extended_enat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

thf(sy_c_Filter_Ocofinite_001t__Nat__Onat,type,
    cofinite_nat: filter_nat ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Int_Oint__ge__less__than,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

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

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

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

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

thf(sy_c_Int_Oring__1__class_OInts_001t__Code____Numeral__Ointeger,type,
    ring_11222124179247155820nteger: set_Code_integer ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
    ring_1_Ints_complex: set_complex ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Int__Oint,type,
    ring_1_Ints_int: set_int ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
    ring_1_Ints_rat: set_rat ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

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

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

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

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

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

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_Eo_J,type,
    inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Int__Oint_M_Eo_J,type,
    inf_inf_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    inf_inf_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
    inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    inf_in5163264567034779214_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Real__Oreal_M_Eo_J,type,
    inf_inf_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
    inf_inf_int: int > int > int ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
    inf_inf_rat: rat > rat > rat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
    inf_inf_real: real > real > real ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
    inf_inf_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    inf_inf_set_complex: set_complex > set_complex > set_complex ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    inf_in8357106775501769908d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
    inf_inf_set_int: set_int > set_int > set_int ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Num__Onum_J,type,
    inf_inf_set_num: set_num > set_num > set_num ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__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_J,type,
    inf_in4302113700860409141at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    inf_in7913087082777306421at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Rat__Orat_J,type,
    inf_inf_set_rat: set_rat > set_rat > set_rat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
    inf_inf_set_real: set_real > set_real > set_real ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

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

thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_M_Eo_J,type,
    sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

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

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    sup_sup_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
    sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__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_M_Eo_J,type,
    sup_su8986005896011022210_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    sup_su2080679670758317954_nat_o: ( produc3843707927480180839at_nat > $o ) > ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Real__Oreal_M_Eo_J,type,
    sup_sup_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

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

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

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
    sup_sup_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    sup_sup_set_complex: set_complex > set_complex > set_complex ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    sup_su4489774667511045786d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).

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

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

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

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

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

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

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

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

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Extended____Nat__Oenat,type,
    lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).

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

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Num__Onum,type,
    lattic8556559851467007577_o_num: ( $o > num ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Rat__Orat,type,
    lattic2140725968369957399_o_rat: ( $o > rat ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Real__Oreal,type,
    lattic8697145971487455083o_real: ( $o > real ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Num__Onum,type,
    lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Num__Onum,type,
    lattic402713867396545063at_num: ( extended_enat > num ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Rat__Orat,type,
    lattic3210252021154270693at_rat: ( extended_enat > rat ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Extended____Nat__Oenat_001t__Real__Oreal,type,
    lattic1189837152898106425t_real: ( extended_enat > real ) > set_Extended_enat > extended_enat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Rat__Orat,type,
    lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
    lattic2675449441010098035t_real: ( int > real ) > set_int > int ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
    lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
    lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
    lattic1613168225601753569al_num: ( real > num ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
    lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
    lattic8440615504127631091l_real: ( real > real ) > set_real > real ).

thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
    lattic7826324295020591184_F_nat: ( nat > nat > nat ) > nat > set_nat > nat ).

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

thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

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

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

thf(sy_c_List_Oconcat_001_Eo,type,
    concat_o: list_list_o > list_o ).

thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

thf(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
    concat_VEBT_VEBT: list_list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Oenumerate_001_Eo,type,
    enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
    enumerate_int: nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
    enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Ofind_001_Eo,type,
    find_o: ( $o > $o ) > list_o > option_o ).

thf(sy_c_List_Ofind_001t__Int__Oint,type,
    find_int: ( int > $o ) > list_int > option_int ).

thf(sy_c_List_Ofind_001t__Nat__Onat,type,
    find_nat: ( nat > $o ) > list_nat > option_nat ).

thf(sy_c_List_Ofind_001t__Num__Onum,type,
    find_num: ( num > $o ) > list_num > option_num ).

thf(sy_c_List_Ofind_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).

thf(sy_c_List_Ofind_001t__Real__Oreal,type,
    find_real: ( real > $o ) > list_real > option_real ).

thf(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
    find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).

thf(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
    find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).

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

thf(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

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

thf(sy_c_List_Olist_OCons_001_Eo,type,
    cons_o: $o > list_o > list_o ).

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

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

thf(sy_c_List_Olist_OCons_001t__Num__Onum,type,
    cons_num: num > list_num > list_num ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

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,
    cons_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
    cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

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

thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
    set_Extended_enat2: list_Extended_enat > set_Extended_enat ).

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
    set_list_o2: list_list_o > set_list_o ).

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT ).

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

thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
    set_num2: list_num > set_num ).

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

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

thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

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

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

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

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

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

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

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

thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
    list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).

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

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

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

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

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

thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_OremoveAll_001_Eo,type,
    removeAll_o: $o > list_o > list_o ).

thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
    removeAll_int: int > list_int > list_int ).

thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
    removeAll_nat: nat > list_nat > list_nat ).

thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
    removeAll_real: real > list_real > list_real ).

thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
    removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
    removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

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

thf(sy_c_List_Orotate1_001_Eo,type,
    rotate1_o: list_o > list_o ).

thf(sy_c_List_Orotate1_001t__Int__Oint,type,
    rotate1_int: list_int > list_int ).

thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
    rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

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

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

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

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

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

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

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

thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
    zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
    zip_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
    zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
    zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    size_s3941691890525107288d_enat: list_Extended_enat > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
    size_s2710708370519433104list_o: list_list_o > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__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__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

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

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

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

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

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

thf(sy_c_Nat__Bijection_Olist__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_NthRoot_Oroot,type,
    root: nat > real > real ).

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

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

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__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_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__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_Opred__numeral,type,
    pred_numeral: num > nat ).

thf(sy_c_Option_Ooption_ONone_001_Eo,type,
    none_o: option_o ).

thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
    none_int: option_int ).

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_ONone_001t__Real__Oreal,type,
    none_real: option_real ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

thf(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
    none_VEBT_VEBT: option_VEBT_VEBT ).

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_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_It__Nat__Onat_J_M_062_It__Option__Ooption_It__Nat__Onat_J_M_Eo_J_J,type,
    bot_bo5043116465536727218_nat_o: option_nat > option_nat > $o ).

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

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

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    bot_bo3364206721330744218_nat_o: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    bot_bo232370072503712749on_nat: set_Pr6588086440996610945on_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_J,type,
    bot_bo5327735625951526323at_nat: set_Pr8693737435421807431at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    bot_bo4948859079157340979at_nat: set_Pr7459493094073627847at_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__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

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__Extended____Nat__Oenat,type,
    ord_Le1955565732374568822d_enat: ( extended_enat > $o ) > extended_enat ).

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

thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
    ord_Least_real: ( real > $o ) > real ).

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $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__List__Olist_It__Nat__Onat_J_J,type,
    ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $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,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Option__Ooption_It__Nat__Onat_J_M_062_It__Option__Ooption_It__Nat__Onat_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_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,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: set_int > set_int > set_int ).

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

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

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

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

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

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

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: ( real > real ) > $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_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_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,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Ounit_ORep__unit,type,
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thf(sy_c_Rat_OAbs__Rat,type,
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thf(sy_c_Rat_OFract,type,
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thf(sy_c_Rat_ORep__Rat,type,
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thf(sy_c_Real_Opcr__real,type,
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thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
    insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat ).

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__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,
    insert5050368324300391991at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

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

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

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

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

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

thf(sy_c_Set_Ois__empty_001_Eo,type,
    is_empty_o: set_o > $o ).

thf(sy_c_Set_Ois__empty_001t__Int__Oint,type,
    is_empty_int: set_int > $o ).

thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
    is_empty_nat: set_nat > $o ).

thf(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
    is_empty_real: set_real > $o ).

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

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

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

thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
    is_sin2641923865335537900st_nat: set_list_nat > $o ).

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

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

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

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
    is_singleton_set_nat: set_set_nat > $o ).

thf(sy_c_Set_Oremove_001_Eo,type,
    remove_o: $o > set_o > set_o ).

thf(sy_c_Set_Oremove_001t__Int__Oint,type,
    remove_int: int > set_int > set_int ).

thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
    remove_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oremove_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    remove6466555014256735590at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
    remove_real: real > set_real > set_real ).

thf(sy_c_Set_Oremove_001t__Set__Oset_It__Nat__Onat_J,type,
    remove_set_nat: set_nat > set_set_nat > set_set_nat ).

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

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

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

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

thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
    the_elem_real: set_real > real ).

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__Int__Oint,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

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

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

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

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
    set_or8904488021354931149Most_o: $o > $o > set_o ).

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

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

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

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

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

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

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_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

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

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

thf(sy_c_Set__Interval_Oord__class_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_001_Eo,type,
    set_ord_lessThan_o: $o > set_o ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Extended____Nat__Oenat,type,
    set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Topological__Spaces_Otopological__space__class_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_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

thf(sy_c_Transcendental_Opowr__real,type,
    powr_real2: real > real > real ).

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

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

thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: complex > complex ).

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

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

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

thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_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_Ogreater__rel,type,
    vEBT_V5711637165310795180er_rel: produc4953844613479565601on_nat > produc4953844613479565601on_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_Oless__rel,type,
    vEBT_VEBT_less_rel: produc4953844613479565601on_nat > produc4953844613479565601on_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_Olesseq__rel,type,
    vEBT_VEBT_lesseq_rel: produc4953844613479565601on_nat > produc4953844613479565601on_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_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_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_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).

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

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

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

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

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    accp_P8646395344606611882on_nat: ( produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ) > produc4953844613479565601on_nat > $o ).

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

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

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
    finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Extended____Nat__Oenat,type,
    finite4251489430341359785d_enat: set_Pr2112562347474612743d_enat ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite469560695537375940at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Ofinite__psubset_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,
    finite4695646753290404266at_nat: set_Pr7459493094073627847at_nat ).

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

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

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

thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Omeasure_001t__Option__Ooption_It__Nat__Onat_J,type,
    measure_option_nat: ( option_nat > nat ) > set_Pr6588086440996610945on_nat ).

thf(sy_c_Wellfounded_Omeasure_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    measur1827424007717751593at_nat: ( set_Pr1261947904930325089at_nat > nat ) > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Omeasure_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    measur4922264856574889999at_nat: ( set_Pr4329608150637261639at_nat > nat ) > set_Pr7459493094073627847at_nat ).

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

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o ).

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

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

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

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

thf(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: extended_enat > set_Extended_enat > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    member4117937158525611210on_nat: produc4953844613479565601on_nat > set_Pr6588086440996610945on_nat > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Extended____Nat__Oenat_J_Mt__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
    member4453595087596390480d_enat: produc1621487020699730983d_enat > set_Pr2112562347474612743d_enat > $o ).

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

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

thf(sy_c_member_001t__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__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

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

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

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

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

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

thf(sy_v_a____,type,
    a: nat ).

thf(sy_v_b____,type,
    b: nat ).

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_h____,type,
    h: nat ).

thf(sy_v_info____,type,
    info: option4927543243414619207at_nat ).

thf(sy_v_k____,type,
    k: vEBT_VEBT ).

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_ta____,type,
    ta: vEBT_VEBT ).

thf(sy_v_treeList_H____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList2: list_VEBT_VEBT ).

% Relevant facts (10123)
thf(fact_0__092_060open_062vebt__mint_At_A_092_060noteq_062_Avebt__mint_Ak_092_060close_062,axiom,
    ( ( vEBT_vebt_mint @ ta )
   != ( vEBT_vebt_mint @ k ) ) ).

% \<open>vebt_mint t \<noteq> vebt_mint k\<close>
thf(fact_1_abdef,axiom,
    ( ( ( ( vEBT_vebt_mint @ ta )
        = none_nat )
      & ( ( vEBT_vebt_mint @ k )
        = ( some_nat @ b ) ) )
    | ( ( ( vEBT_vebt_mint @ ta )
        = ( some_nat @ a ) )
      & ( ( vEBT_vebt_mint @ k )
        = none_nat ) )
    | ( ( ord_less_nat @ a @ b )
      & ( ( some_nat @ a )
        = ( vEBT_vebt_mint @ ta ) )
      & ( ( some_nat @ b )
        = ( vEBT_vebt_mint @ k ) ) )
    | ( ( ord_less_nat @ b @ a )
      & ( ( some_nat @ a )
        = ( vEBT_vebt_mint @ ta ) )
      & ( ( some_nat @ b )
        = ( vEBT_vebt_mint @ k ) ) ) ) ).

% abdef
thf(fact_2_assms_I3_J,axiom,
    ( ( vEBT_VEBT_set_vebt @ ta )
    = ( vEBT_VEBT_set_vebt @ k ) ) ).

% assms(3)
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_Ab_O_Avebt__mint_At_A_061_ANone_A_092_060and_062_Avebt__mint_Ak_A_061_ASome_Ab_A_092_060or_062_Avebt__mint_At_A_061_ASome_Aa_A_092_060and_062_Avebt__mint_Ak_A_061_ANone_A_092_060or_062_Aa_A_060_Ab_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060or_062_Ab_A_060_Aa_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [A: nat,B: nat] :
        ~ ( ( ( ( vEBT_vebt_mint @ ta )
              = none_nat )
            & ( ( vEBT_vebt_mint @ k )
              = ( some_nat @ B ) ) )
          | ( ( ( vEBT_vebt_mint @ ta )
              = ( some_nat @ A ) )
            & ( ( vEBT_vebt_mint @ k )
              = none_nat ) )
          | ( ( ord_less_nat @ A @ B )
            & ( ( some_nat @ A )
              = ( vEBT_vebt_mint @ ta ) )
            & ( ( some_nat @ B )
              = ( vEBT_vebt_mint @ k ) ) )
          | ( ( ord_less_nat @ B @ A )
            & ( ( some_nat @ A )
              = ( vEBT_vebt_mint @ ta ) )
            & ( ( some_nat @ B )
              = ( vEBT_vebt_mint @ k ) ) ) ) ).

% \<open>\<And>thesis. (\<And>a b. vebt_mint t = None \<and> vebt_mint k = Some b \<or> vebt_mint t = Some a \<and> vebt_mint k = None \<or> a < b \<and> Some a = vebt_mint t \<and> Some b = vebt_mint k \<or> b < a \<and> Some a = vebt_mint t \<and> Some b = vebt_mint k \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_5_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_6_not__None__eq,axiom,
    ! [X2: option_nat] :
      ( ( X2 != none_nat )
      = ( ? [Y: nat] :
            ( X2
            = ( some_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_7_not__None__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( X2 != none_P5556105721700978146at_nat )
      = ( ? [Y: product_prod_nat_nat] :
            ( X2
            = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_8_not__None__eq,axiom,
    ! [X2: option_num] :
      ( ( X2 != none_num )
      = ( ? [Y: num] :
            ( X2
            = ( some_num @ Y ) ) ) ) ).

% not_None_eq
thf(fact_9_not__Some__eq,axiom,
    ! [X2: option_nat] :
      ( ( ! [Y: nat] :
            ( X2
           != ( some_nat @ Y ) ) )
      = ( X2 = none_nat ) ) ).

% not_Some_eq
thf(fact_10_not__Some__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( ! [Y: product_prod_nat_nat] :
            ( X2
           != ( some_P7363390416028606310at_nat @ Y ) ) )
      = ( X2 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_11_not__Some__eq,axiom,
    ! [X2: option_num] :
      ( ( ! [Y: num] :
            ( X2
           != ( some_num @ Y ) ) )
      = ( X2 = none_num ) ) ).

% not_Some_eq
thf(fact_12_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_13_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_14_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_15_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_16_option_Oinject,axiom,
    ! [X22: num,Y2: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_17_assms_I2_J,axiom,
    vEBT_invar_vebt @ k @ h ).

% assms(2)
thf(fact_18_assms_I1_J,axiom,
    vEBT_invar_vebt @ ta @ h ).

% assms(1)
thf(fact_19_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_20_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_21_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_22_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_23_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_24_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_25_option_Oexhaust,axiom,
    ! [Y3: option_nat] :
      ( ( Y3 != none_nat )
     => ~ ! [X23: nat] :
            ( Y3
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_26_option_Oexhaust,axiom,
    ! [Y3: option4927543243414619207at_nat] :
      ( ( Y3 != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y3
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_27_option_Oexhaust,axiom,
    ! [Y3: option_num] :
      ( ( Y3 != none_num )
     => ~ ! [X23: num] :
            ( Y3
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_28_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X2 ) @ N ) ) ).

% insert'_pres_valid
thf(fact_29_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X2 ) ) ) ) ).

% mint_sound
thf(fact_30_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).

% mint_corr
thf(fact_31_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_32_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: nat,B: nat] :
              ( ( X2
                = ( some_nat @ A ) )
             => ( ( Y3
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_33_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: nat,B: product_prod_nat_nat] :
              ( ( X2
                = ( some_nat @ A ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_34_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: nat,B: num] :
              ( ( X2
                = ( some_nat @ A ) )
             => ( ( Y3
                  = ( some_num @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_35_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: product_prod_nat_nat,B: nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A ) )
             => ( ( Y3
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_36_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_37_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: product_prod_nat_nat,B: num] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A ) )
             => ( ( Y3
                  = ( some_num @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_38_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: num,B: nat] :
              ( ( X2
                = ( some_num @ A ) )
             => ( ( Y3
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_39_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: num,B: product_prod_nat_nat] :
              ( ( X2
                = ( some_num @ A ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_40_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_num > $o,Y3: option_num] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A: num,B: num] :
              ( ( X2
                = ( some_num @ A ) )
             => ( ( Y3
                  = ( some_num @ B ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_41_split__option__all,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ! [X3: option_nat] : ( P2 @ X3 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_42_split__option__all,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ! [X3: option4927543243414619207at_nat] : ( P2 @ X3 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_43_split__option__all,axiom,
    ( ( ^ [P2: option_num > $o] :
        ! [X3: option_num] : ( P2 @ X3 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_44_split__option__ex,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ? [X3: option_nat] : ( P2 @ X3 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_45_split__option__ex,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ? [X3: option4927543243414619207at_nat] : ( P2 @ X3 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_46_split__option__ex,axiom,
    ( ( ^ [P2: option_num > $o] :
        ? [X3: option_num] : ( P2 @ X3 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_47_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_48_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_49_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_50_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_51_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_52_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_53_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_54_case4_I12_J,axiom,
    vEBT_invar_vebt @ sa @ deg ).

% case4(12)
thf(fact_55_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% succ_corr
thf(fact_56_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).

% pred_corr
thf(fact_57_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_58_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_59_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).

% min_Null_member
thf(fact_60_pred__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A2 @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs )
              & ( ord_less_nat @ X4 @ A2 ) ) ) ) ).

% pred_none_empty
thf(fact_61_succ__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A2 @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs )
              & ( ord_less_nat @ A2 @ X4 ) ) ) ) ).

% succ_none_empty
thf(fact_62_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X2 )
        = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_63_obtain__set__pred,axiom,
    ! [Z: nat,X2: nat,A3: set_nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
       => ( ( finite_finite_nat @ A3 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A3 @ X2 @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_64_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% pred_correct
thf(fact_65_mem__Collect__eq,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( member_o @ A2 @ ( collect_o @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A2: real,P: real > $o] :
      ( ( member_real @ A2 @ ( collect_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A2: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
    ! [A2: int,P: int > $o] :
      ( ( member_int @ A2 @ ( collect_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A3: set_o] :
      ( ( collect_o
        @ ^ [X: $o] : ( member_o @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A3: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( member_set_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_77_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_78_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_79_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_80_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_81_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_82_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).

% succ_correct
thf(fact_83_obtain__set__succ,axiom,
    ! [X2: nat,Z: nat,A3: set_nat,B2: set_nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
       => ( ( finite_finite_nat @ B2 )
         => ( ( A3 = B2 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X2 @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_84_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_85_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_86_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_87_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_88_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_89_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_90_succ__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
      = ( ( vEBT_vebt_member @ T @ Y3 )
        & ( ord_less_nat @ X2 @ Y3 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ X2 @ Z2 ) )
           => ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ) ).

% succ_member
thf(fact_91_pred__member,axiom,
    ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
      = ( ( vEBT_vebt_member @ T @ Y3 )
        & ( ord_less_nat @ Y3 @ X2 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ Z2 @ X2 ) )
           => ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ).

% pred_member
thf(fact_92_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X2 )
         => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).

% mint_corr_help
thf(fact_93_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_94_ex__min__if__finite,axiom,
    ! [S: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ S )
            & ~ ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ S )
                  & ( ord_le72135733267957522d_enat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_95_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ S )
            & ~ ? [Xa: $o] :
                  ( ( member_o @ Xa @ S )
                  & ( ord_less_o @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_96_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ S )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_97_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( ( finite_finite_rat @ S )
     => ( ( S != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ S )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S )
                  & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_98_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( ( finite_finite_num @ S )
     => ( ( S != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ S )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S )
                  & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_99_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ S )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_100_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ S )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_101_infinite__growing,axiom,
    ! [X6: set_Extended_enat] :
      ( ( X6 != bot_bo7653980558646680370d_enat )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ X6 )
           => ? [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ X6 )
                & ( ord_le72135733267957522d_enat @ X5 @ Xa ) ) )
       => ~ ( finite4001608067531595151d_enat @ X6 ) ) ) ).

% infinite_growing
thf(fact_102_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ X6 )
           => ? [Xa: $o] :
                ( ( member_o @ Xa @ X6 )
                & ( ord_less_o @ X5 @ Xa ) ) )
       => ~ ( finite_finite_o @ X6 ) ) ) ).

% infinite_growing
thf(fact_103_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ X6 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X6 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X6 ) ) ) ).

% infinite_growing
thf(fact_104_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( ( member_rat @ X5 @ X6 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X6 )
                & ( ord_less_rat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X6 ) ) ) ).

% infinite_growing
thf(fact_105_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( ( member_num @ X5 @ X6 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X6 )
                & ( ord_less_num @ X5 @ Xa ) ) )
       => ~ ( finite_finite_num @ X6 ) ) ) ).

% infinite_growing
thf(fact_106_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ X6 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X6 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_107_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ X6 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X6 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X6 ) ) ) ).

% infinite_growing
thf(fact_108_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs2: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs2 )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs2 )
             => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_109_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs2: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs2 )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs2 )
             => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_110_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_111_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X2 )
         => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_112_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).

% maxt_corr
thf(fact_113_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X2 ) ) ) ) ).

% maxt_sound
thf(fact_114_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_115_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_116_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_117_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_118_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_119_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_120_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_121_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_122_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_123_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B3 ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_124_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_125_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_126_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_127_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_128_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_129_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_130_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_131_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_132_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_nat @ M2 @ N2 )
          | ( M2 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_133_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_134_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( M2 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_135_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_136_zero__reorient,axiom,
    ! [X2: literal] :
      ( ( zero_zero_literal = X2 )
      = ( X2 = zero_zero_literal ) ) ).

% zero_reorient
thf(fact_137_zero__reorient,axiom,
    ! [X2: real] :
      ( ( zero_zero_real = X2 )
      = ( X2 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_138_zero__reorient,axiom,
    ! [X2: rat] :
      ( ( zero_zero_rat = X2 )
      = ( X2 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_139_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_140_zero__reorient,axiom,
    ! [X2: int] :
      ( ( zero_zero_int = X2 )
      = ( X2 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_141_linorder__neqE__nat,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_142_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_143_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_144_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_145_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_146_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_147_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_148_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_149_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_150_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_151_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_152_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_153_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N3 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_154_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_155_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_156_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_157_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_158_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_159_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_160_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).

% buildup_nothing_in_leaf
thf(fact_161_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs2: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs2 )
          & ( ord_less_nat @ X @ Y )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs2 )
             => ( ( ord_less_nat @ X @ Z2 )
               => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_162_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs2: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs2 )
          & ( ord_less_nat @ Y @ X )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs2 )
             => ( ( ord_less_nat @ Z2 @ X )
               => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_163_finite__has__maximal,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_164_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_165_finite__has__maximal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A3 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A3 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_166_finite__has__maximal,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A3 )
               => ( ( ord_less_eq_set_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_167_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_168_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_169_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_170_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_171_finite__has__minimal,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_172_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_173_finite__has__minimal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A3 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A3 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_174_finite__has__minimal,axiom,
    ! [A3: set_set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A3 )
               => ( ( ord_less_eq_set_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_175_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( A3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_176_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( ( finite_finite_num @ A3 )
     => ( ( A3 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_177_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_178_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_179_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_180_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_181_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_182_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_183_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_184_all__not__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ! [X: set_nat] :
            ~ ( member_set_nat @ X @ A3 ) )
      = ( A3 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_185_all__not__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ! [X: real] :
            ~ ( member_real @ X @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_186_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ! [X: $o] :
            ~ ( member_o @ X @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_187_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_188_all__not__in__conv,axiom,
    ! [A3: set_int] :
      ( ( ! [X: int] :
            ~ ( member_int @ X @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_189_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X: list_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_190_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_191_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_192_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_193_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_194_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X: int] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_195_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X: list_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_196_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_197_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_198_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_199_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_200_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X: int] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_201_order__refl,axiom,
    ! [X2: set_int] : ( ord_less_eq_set_int @ X2 @ X2 ) ).

% order_refl
thf(fact_202_order__refl,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).

% order_refl
thf(fact_203_order__refl,axiom,
    ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).

% order_refl
thf(fact_204_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_205_order__refl,axiom,
    ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).

% order_refl
thf(fact_206_dual__order_Orefl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_207_dual__order_Orefl,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_208_dual__order_Orefl,axiom,
    ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_209_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_210_dual__order_Orefl,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_211_empty__subsetI,axiom,
    ! [A3: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A3 ) ).

% empty_subsetI
thf(fact_212_empty__subsetI,axiom,
    ! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).

% empty_subsetI
thf(fact_213_empty__subsetI,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).

% empty_subsetI
thf(fact_214_empty__subsetI,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).

% empty_subsetI
thf(fact_215_subset__empty,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ bot_bot_set_real )
      = ( A3 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_216_subset__empty,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_217_subset__empty,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_218_subset__empty,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_219_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_220_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_221_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_222_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_223_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_224_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_225_rev__finite__subset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_226_rev__finite__subset,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( finite3207457112153483333omplex @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_227_rev__finite__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
       => ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_228_rev__finite__subset,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( finite4001608067531595151d_enat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_229_rev__finite__subset,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A3 @ B2 )
       => ( finite_finite_int @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_230_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_231_infinite__super,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ( ord_le211207098394363844omplex @ S @ T2 )
     => ( ~ ( finite3207457112153483333omplex @ S )
       => ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).

% infinite_super
thf(fact_232_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T2 )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_233_infinite__super,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ S @ T2 )
     => ( ~ ( finite4001608067531595151d_enat @ S )
       => ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).

% infinite_super
thf(fact_234_infinite__super,axiom,
    ! [S: set_int,T2: set_int] :
      ( ( ord_less_eq_set_int @ S @ T2 )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ T2 ) ) ) ).

% infinite_super
thf(fact_235_finite__subset,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_236_finite__subset,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A3 @ B2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( finite3207457112153483333omplex @ A3 ) ) ) ).

% finite_subset
thf(fact_237_finite__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_238_finite__subset,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( finite4001608067531595151d_enat @ A3 ) ) ) ).

% finite_subset
thf(fact_239_finite__subset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( finite_finite_int @ B2 )
       => ( finite_finite_int @ A3 ) ) ) ).

% finite_subset
thf(fact_240_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_241_order__antisym__conv,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X2 )
     => ( ( ord_less_eq_set_int @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_242_order__antisym__conv,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_243_order__antisym__conv,axiom,
    ! [Y3: num,X2: num] :
      ( ( ord_less_eq_num @ Y3 @ X2 )
     => ( ( ord_less_eq_num @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_244_order__antisym__conv,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_245_order__antisym__conv,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ Y3 @ X2 )
     => ( ( ord_less_eq_int @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_246_linorder__le__cases,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).

% linorder_le_cases
thf(fact_247_linorder__le__cases,axiom,
    ! [X2: num,Y3: num] :
      ( ~ ( ord_less_eq_num @ X2 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X2 ) ) ).

% linorder_le_cases
thf(fact_248_linorder__le__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_le_cases
thf(fact_249_linorder__le__cases,axiom,
    ! [X2: int,Y3: int] :
      ( ~ ( ord_less_eq_int @ X2 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X2 ) ) ).

% linorder_le_cases
thf(fact_250_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_252_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_253_ord__le__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_254_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_255_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_256_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_257_ord__le__eq__subst,axiom,
    ! [A2: num,B3: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_258_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_259_ord__le__eq__subst,axiom,
    ! [A2: nat,B3: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_260_ord__eq__le__subst,axiom,
    ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_261_ord__eq__le__subst,axiom,
    ! [A2: num,F: rat > num,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_262_ord__eq__le__subst,axiom,
    ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_263_ord__eq__le__subst,axiom,
    ! [A2: int,F: rat > int,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_264_ord__eq__le__subst,axiom,
    ! [A2: rat,F: num > rat,B3: num,C: num] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_265_ord__eq__le__subst,axiom,
    ! [A2: num,F: num > num,B3: num,C: num] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_266_ord__eq__le__subst,axiom,
    ! [A2: nat,F: num > nat,B3: num,C: num] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_267_ord__eq__le__subst,axiom,
    ! [A2: int,F: num > int,B3: num,C: num] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_268_ord__eq__le__subst,axiom,
    ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_269_ord__eq__le__subst,axiom,
    ! [A2: num,F: nat > num,B3: nat,C: nat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_270_linorder__linear,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
      | ( ord_less_eq_rat @ Y3 @ X2 ) ) ).

% linorder_linear
thf(fact_271_linorder__linear,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
      | ( ord_less_eq_num @ Y3 @ X2 ) ) ).

% linorder_linear
thf(fact_272_linorder__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_linear
thf(fact_273_linorder__linear,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
      | ( ord_less_eq_int @ Y3 @ X2 ) ) ).

% linorder_linear
thf(fact_274_order__eq__refl,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( X2 = Y3 )
     => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).

% order_eq_refl
thf(fact_275_order__eq__refl,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 = Y3 )
     => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).

% order_eq_refl
thf(fact_276_order__eq__refl,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 = Y3 )
     => ( ord_less_eq_num @ X2 @ Y3 ) ) ).

% order_eq_refl
thf(fact_277_order__eq__refl,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 = Y3 )
     => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).

% order_eq_refl
thf(fact_278_order__eq__refl,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 = Y3 )
     => ( ord_less_eq_int @ X2 @ Y3 ) ) ).

% order_eq_refl
thf(fact_279_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_280_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_281_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_282_order__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_283_order__subst2,axiom,
    ! [A2: num,B3: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_284_order__subst2,axiom,
    ! [A2: num,B3: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_285_order__subst2,axiom,
    ! [A2: num,B3: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_286_order__subst2,axiom,
    ! [A2: num,B3: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_287_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_288_order__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_289_order__subst1,axiom,
    ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_290_order__subst1,axiom,
    ! [A2: rat,F: num > rat,B3: num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_291_order__subst1,axiom,
    ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_292_order__subst1,axiom,
    ! [A2: rat,F: int > rat,B3: int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_eq_int @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_293_order__subst1,axiom,
    ! [A2: num,F: rat > num,B3: rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_294_order__subst1,axiom,
    ! [A2: num,F: num > num,B3: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_295_order__subst1,axiom,
    ! [A2: num,F: nat > num,B3: nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_296_order__subst1,axiom,
    ! [A2: num,F: int > num,B3: int,C: int] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_eq_int @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_297_order__subst1,axiom,
    ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_298_order__subst1,axiom,
    ! [A2: nat,F: num > nat,B3: num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_299_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z3: set_int] : Y6 = Z3 )
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_300_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_301_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z3: num] : Y6 = Z3 )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_302_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_303_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_304_antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_305_antisym,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_306_antisym,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_eq_num @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_307_antisym,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_308_antisym,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% antisym
thf(fact_309_dual__order_Otrans,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( ord_less_eq_set_int @ C @ B3 )
       => ( ord_less_eq_set_int @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_310_dual__order_Otrans,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ C @ B3 )
       => ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_311_dual__order_Otrans,axiom,
    ! [B3: num,A2: num,C: num] :
      ( ( ord_less_eq_num @ B3 @ A2 )
     => ( ( ord_less_eq_num @ C @ B3 )
       => ( ord_less_eq_num @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_312_dual__order_Otrans,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_313_dual__order_Otrans,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ C @ B3 )
       => ( ord_less_eq_int @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_314_dual__order_Oantisym,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_315_dual__order_Oantisym,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_316_dual__order_Oantisym,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_eq_num @ B3 @ A2 )
     => ( ( ord_less_eq_num @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_317_dual__order_Oantisym,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_318_dual__order_Oantisym,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ A2 @ B3 )
       => ( A2 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_319_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_int,Z3: set_int] : Y6 = Z3 )
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A4 )
          & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_320_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_321_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z3: num] : Y6 = Z3 )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_322_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_323_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_324_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A2: rat,B3: rat] :
      ( ! [A: rat,B: rat] :
          ( ( ord_less_eq_rat @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: rat,B: rat] :
            ( ( P @ B @ A )
           => ( P @ A @ B ) )
       => ( P @ A2 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_325_linorder__wlog,axiom,
    ! [P: num > num > $o,A2: num,B3: num] :
      ( ! [A: num,B: num] :
          ( ( ord_less_eq_num @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: num,B: num] :
            ( ( P @ B @ A )
           => ( P @ A @ B ) )
       => ( P @ A2 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_326_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: nat,B: nat] :
            ( ( P @ B @ A )
           => ( P @ A @ B ) )
       => ( P @ A2 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_327_linorder__wlog,axiom,
    ! [P: int > int > $o,A2: int,B3: int] :
      ( ! [A: int,B: int] :
          ( ( ord_less_eq_int @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: int,B: int] :
            ( ( P @ B @ A )
           => ( P @ A @ B ) )
       => ( P @ A2 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_328_order__trans,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ Z )
       => ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_329_order__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z )
       => ( ord_less_eq_rat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_330_order__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z )
       => ( ord_less_eq_num @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_331_order__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_332_order__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_333_order_Otrans,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C )
       => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).

% order.trans
thf(fact_334_order_Otrans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_335_order_Otrans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ord_less_eq_num @ A2 @ C ) ) ) ).

% order.trans
thf(fact_336_order_Otrans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_337_order_Otrans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% order.trans
thf(fact_338_order__antisym,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ X2 )
       => ( X2 = Y3 ) ) ) ).

% order_antisym
thf(fact_339_order__antisym,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ X2 )
       => ( X2 = Y3 ) ) ) ).

% order_antisym
thf(fact_340_order__antisym,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ X2 )
       => ( X2 = Y3 ) ) ) ).

% order_antisym
thf(fact_341_order__antisym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X2 )
       => ( X2 = Y3 ) ) ) ).

% order_antisym
thf(fact_342_order__antisym,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ X2 )
       => ( X2 = Y3 ) ) ) ).

% order_antisym
thf(fact_343_ord__le__eq__trans,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_344_ord__le__eq__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_345_ord__le__eq__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_num @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_346_ord__le__eq__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_347_ord__le__eq__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_348_ord__eq__le__trans,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( A2 = B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C )
       => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_349_ord__eq__le__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2 = B3 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_350_ord__eq__le__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( A2 = B3 )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ord_less_eq_num @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_351_ord__eq__le__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( A2 = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_352_ord__eq__le__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( A2 = B3 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_353_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z3: set_int] : Y6 = Z3 )
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_354_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_355_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z3: num] : Y6 = Z3 )
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_356_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_357_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_358_le__cases3,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X2 @ Y3 )
       => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y3 @ X2 )
         => ~ ( ord_less_eq_rat @ X2 @ Z ) )
       => ( ( ( ord_less_eq_rat @ X2 @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y3 )
             => ~ ( ord_less_eq_rat @ Y3 @ X2 ) )
           => ( ( ( ord_less_eq_rat @ Y3 @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X2 )
                 => ~ ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_359_le__cases3,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( ( ord_less_eq_num @ X2 @ Y3 )
       => ~ ( ord_less_eq_num @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_num @ Y3 @ X2 )
         => ~ ( ord_less_eq_num @ X2 @ Z ) )
       => ( ( ( ord_less_eq_num @ X2 @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_num @ Z @ Y3 )
             => ~ ( ord_less_eq_num @ Y3 @ X2 ) )
           => ( ( ( ord_less_eq_num @ Y3 @ Z )
               => ~ ( ord_less_eq_num @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_num @ Z @ X2 )
                 => ~ ( ord_less_eq_num @ X2 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_360_le__cases3,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_361_le__cases3,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ( ord_less_eq_int @ X2 @ Y3 )
       => ~ ( ord_less_eq_int @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_int @ Y3 @ X2 )
         => ~ ( ord_less_eq_int @ X2 @ Z ) )
       => ( ( ( ord_less_eq_int @ X2 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_int @ Z @ Y3 )
             => ~ ( ord_less_eq_int @ Y3 @ X2 ) )
           => ( ( ( ord_less_eq_int @ Y3 @ Z )
               => ~ ( ord_less_eq_int @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X2 )
                 => ~ ( ord_less_eq_int @ X2 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_362_nle__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ~ ( ord_less_eq_rat @ A2 @ B3 ) )
      = ( ( ord_less_eq_rat @ B3 @ A2 )
        & ( B3 != A2 ) ) ) ).

% nle_le
thf(fact_363_nle__le,axiom,
    ! [A2: num,B3: num] :
      ( ( ~ ( ord_less_eq_num @ A2 @ B3 ) )
      = ( ( ord_less_eq_num @ B3 @ A2 )
        & ( B3 != A2 ) ) ) ).

% nle_le
thf(fact_364_nle__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A2 )
        & ( B3 != A2 ) ) ) ).

% nle_le
thf(fact_365_nle__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ~ ( ord_less_eq_int @ A2 @ B3 ) )
      = ( ( ord_less_eq_int @ B3 @ A2 )
        & ( B3 != A2 ) ) ) ).

% nle_le
thf(fact_366_lt__ex,axiom,
    ! [X2: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).

% lt_ex
thf(fact_367_lt__ex,axiom,
    ! [X2: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).

% lt_ex
thf(fact_368_lt__ex,axiom,
    ! [X2: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).

% lt_ex
thf(fact_369_gt__ex,axiom,
    ! [X2: real] :
    ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).

% gt_ex
thf(fact_370_gt__ex,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_371_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_372_gt__ex,axiom,
    ! [X2: int] :
    ? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).

% gt_ex
thf(fact_373_dense,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ? [Z4: real] :
          ( ( ord_less_real @ X2 @ Z4 )
          & ( ord_less_real @ Z4 @ Y3 ) ) ) ).

% dense
thf(fact_374_dense,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ? [Z4: rat] :
          ( ( ord_less_rat @ X2 @ Z4 )
          & ( ord_less_rat @ Z4 @ Y3 ) ) ) ).

% dense
thf(fact_375_less__imp__neq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_376_less__imp__neq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_377_less__imp__neq,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_378_less__imp__neq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_379_less__imp__neq,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_380_order_Oasym,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ~ ( ord_less_real @ B3 @ A2 ) ) ).

% order.asym
thf(fact_381_order_Oasym,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ~ ( ord_less_rat @ B3 @ A2 ) ) ).

% order.asym
thf(fact_382_order_Oasym,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ~ ( ord_less_num @ B3 @ A2 ) ) ).

% order.asym
thf(fact_383_order_Oasym,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A2 ) ) ).

% order.asym
thf(fact_384_order_Oasym,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ~ ( ord_less_int @ B3 @ A2 ) ) ).

% order.asym
thf(fact_385_ord__eq__less__trans,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( A2 = B3 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_386_ord__eq__less__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2 = B3 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_387_ord__eq__less__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( A2 = B3 )
     => ( ( ord_less_num @ B3 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_388_ord__eq__less__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( A2 = B3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_389_ord__eq__less__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( A2 = B3 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_390_ord__less__eq__trans,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_391_ord__less__eq__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_392_ord__less__eq__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_393_ord__less__eq__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_394_ord__less__eq__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( B3 = C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_395_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X5: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X5 )
             => ( P @ Y5 ) )
         => ( P @ X5 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_396_antisym__conv3,axiom,
    ! [Y3: real,X2: real] :
      ( ~ ( ord_less_real @ Y3 @ X2 )
     => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_397_antisym__conv3,axiom,
    ! [Y3: rat,X2: rat] :
      ( ~ ( ord_less_rat @ Y3 @ X2 )
     => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_398_antisym__conv3,axiom,
    ! [Y3: num,X2: num] :
      ( ~ ( ord_less_num @ Y3 @ X2 )
     => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_399_antisym__conv3,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_400_antisym__conv3,axiom,
    ! [Y3: int,X2: int] :
      ( ~ ( ord_less_int @ Y3 @ X2 )
     => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_401_linorder__cases,axiom,
    ! [X2: real,Y3: real] :
      ( ~ ( ord_less_real @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_real @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_402_linorder__cases,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_rat @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_403_linorder__cases,axiom,
    ! [X2: num,Y3: num] :
      ( ~ ( ord_less_num @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_num @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_404_linorder__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_405_linorder__cases,axiom,
    ! [X2: int,Y3: int] :
      ( ~ ( ord_less_int @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_int @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_406_dual__order_Oasym,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ~ ( ord_less_real @ A2 @ B3 ) ) ).

% dual_order.asym
thf(fact_407_dual__order_Oasym,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ~ ( ord_less_rat @ A2 @ B3 ) ) ).

% dual_order.asym
thf(fact_408_dual__order_Oasym,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ~ ( ord_less_num @ A2 @ B3 ) ) ).

% dual_order.asym
thf(fact_409_dual__order_Oasym,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B3 ) ) ).

% dual_order.asym
thf(fact_410_dual__order_Oasym,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ~ ( ord_less_int @ A2 @ B3 ) ) ).

% dual_order.asym
thf(fact_411_dual__order_Oirrefl,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_412_dual__order_Oirrefl,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_413_dual__order_Oirrefl,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_414_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_415_dual__order_Oirrefl,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_416_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X3: nat] : ( P2 @ X3 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ~ ( P3 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_417_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A2: real,B3: real] :
      ( ! [A: real,B: real] :
          ( ( ord_less_real @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: real] : ( P @ A @ A )
       => ( ! [A: real,B: real] :
              ( ( P @ B @ A )
             => ( P @ A @ B ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_418_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A2: rat,B3: rat] :
      ( ! [A: rat,B: rat] :
          ( ( ord_less_rat @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: rat] : ( P @ A @ A )
       => ( ! [A: rat,B: rat] :
              ( ( P @ B @ A )
             => ( P @ A @ B ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_419_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A2: num,B3: num] :
      ( ! [A: num,B: num] :
          ( ( ord_less_num @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: num] : ( P @ A @ A )
       => ( ! [A: num,B: num] :
              ( ( P @ B @ A )
             => ( P @ A @ B ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_420_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( ord_less_nat @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: nat] : ( P @ A @ A )
       => ( ! [A: nat,B: nat] :
              ( ( P @ B @ A )
             => ( P @ A @ B ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_421_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A2: int,B3: int] :
      ( ! [A: int,B: int] :
          ( ( ord_less_int @ A @ B )
         => ( P @ A @ B ) )
     => ( ! [A: int] : ( P @ A @ A )
       => ( ! [A: int,B: int] :
              ( ( P @ B @ A )
             => ( P @ A @ B ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_422_order_Ostrict__trans,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_423_order_Ostrict__trans,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_424_order_Ostrict__trans,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ( ord_less_num @ B3 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_425_order_Ostrict__trans,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_426_order_Ostrict__trans,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_427_not__less__iff__gr__or__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
      = ( ( ord_less_real @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_428_not__less__iff__gr__or__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
      = ( ( ord_less_rat @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_429_not__less__iff__gr__or__eq,axiom,
    ! [X2: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
      = ( ( ord_less_num @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_430_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_431_not__less__iff__gr__or__eq,axiom,
    ! [X2: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
      = ( ( ord_less_int @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_432_dual__order_Ostrict__trans,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_less_real @ C @ B3 )
       => ( ord_less_real @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_433_dual__order_Ostrict__trans,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_less_rat @ C @ B3 )
       => ( ord_less_rat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_434_dual__order_Ostrict__trans,axiom,
    ! [B3: num,A2: num,C: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ( ( ord_less_num @ C @ B3 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_435_dual__order_Ostrict__trans,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( ord_less_nat @ C @ B3 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_436_dual__order_Ostrict__trans,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( ord_less_int @ C @ B3 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_437_order_Ostrict__implies__not__eq,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_438_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_439_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_440_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_441_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( A2 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_442_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_443_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_444_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_445_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_446_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( A2 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_447_linorder__neqE,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_real @ X2 @ Y3 )
       => ( ord_less_real @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_448_linorder__neqE,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_rat @ X2 @ Y3 )
       => ( ord_less_rat @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_449_linorder__neqE,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_num @ X2 @ Y3 )
       => ( ord_less_num @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_450_linorder__neqE,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_451_linorder__neqE,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_int @ X2 @ Y3 )
       => ( ord_less_int @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_452_order__less__asym,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_453_order__less__asym,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_454_order__less__asym,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_455_order__less__asym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_456_order__less__asym,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_457_linorder__neq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
      = ( ( ord_less_real @ X2 @ Y3 )
        | ( ord_less_real @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_458_linorder__neq__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 != Y3 )
      = ( ( ord_less_rat @ X2 @ Y3 )
        | ( ord_less_rat @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_459_linorder__neq__iff,axiom,
    ! [X2: num,Y3: num] :
      ( ( X2 != Y3 )
      = ( ( ord_less_num @ X2 @ Y3 )
        | ( ord_less_num @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_460_linorder__neq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
      = ( ( ord_less_nat @ X2 @ Y3 )
        | ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_461_linorder__neq__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
      = ( ( ord_less_int @ X2 @ Y3 )
        | ( ord_less_int @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_462_order__less__asym_H,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ~ ( ord_less_real @ B3 @ A2 ) ) ).

% order_less_asym'
thf(fact_463_order__less__asym_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ~ ( ord_less_rat @ B3 @ A2 ) ) ).

% order_less_asym'
thf(fact_464_order__less__asym_H,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ~ ( ord_less_num @ B3 @ A2 ) ) ).

% order_less_asym'
thf(fact_465_order__less__asym_H,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A2 ) ) ).

% order_less_asym'
thf(fact_466_order__less__asym_H,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ~ ( ord_less_int @ B3 @ A2 ) ) ).

% order_less_asym'
thf(fact_467_order__less__trans,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z )
       => ( ord_less_real @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_468_order__less__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z )
       => ( ord_less_rat @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_469_order__less__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z )
       => ( ord_less_num @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_470_order__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_471_order__less__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_472_ord__eq__less__subst,axiom,
    ! [A2: real,F: real > real,B3: real,C: real] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_473_ord__eq__less__subst,axiom,
    ! [A2: rat,F: real > rat,B3: real,C: real] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_474_ord__eq__less__subst,axiom,
    ! [A2: num,F: real > num,B3: real,C: real] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_475_ord__eq__less__subst,axiom,
    ! [A2: nat,F: real > nat,B3: real,C: real] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_476_ord__eq__less__subst,axiom,
    ! [A2: int,F: real > int,B3: real,C: real] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_477_ord__eq__less__subst,axiom,
    ! [A2: real,F: rat > real,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_478_ord__eq__less__subst,axiom,
    ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_479_ord__eq__less__subst,axiom,
    ! [A2: num,F: rat > num,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_480_ord__eq__less__subst,axiom,
    ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_481_ord__eq__less__subst,axiom,
    ! [A2: int,F: rat > int,B3: rat,C: rat] :
      ( ( A2
        = ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_482_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: real > real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_483_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_484_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: real > num,C: num] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_485_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_486_ord__less__eq__subst,axiom,
    ! [A2: real,B3: real,F: real > int,C: int] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_487_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_488_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_489_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_490_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_491_ord__less__eq__subst,axiom,
    ! [A2: rat,B3: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ( F @ B3 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_492_order__less__irrefl,axiom,
    ! [X2: real] :
      ~ ( ord_less_real @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_493_order__less__irrefl,axiom,
    ! [X2: rat] :
      ~ ( ord_less_rat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_494_order__less__irrefl,axiom,
    ! [X2: num] :
      ~ ( ord_less_num @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_495_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_496_order__less__irrefl,axiom,
    ! [X2: int] :
      ~ ( ord_less_int @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_497_order__less__subst1,axiom,
    ! [A2: real,F: real > real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_498_order__less__subst1,axiom,
    ! [A2: real,F: rat > real,B3: rat,C: rat] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_499_order__less__subst1,axiom,
    ! [A2: real,F: num > real,B3: num,C: num] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_500_order__less__subst1,axiom,
    ! [A2: real,F: nat > real,B3: nat,C: nat] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_501_order__less__subst1,axiom,
    ! [A2: real,F: int > real,B3: int,C: int] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_int @ B3 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_502_order__less__subst1,axiom,
    ! [A2: rat,F: real > rat,B3: real,C: real] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_503_order__less__subst1,axiom,
    ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_504_order__less__subst1,axiom,
    ! [A2: rat,F: num > rat,B3: num,C: num] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_505_order__less__subst1,axiom,
    ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_506_order__less__subst1,axiom,
    ! [A2: rat,F: int > rat,B3: int,C: int] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_int @ B3 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_507_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: real > real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_508_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_509_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: real > num,C: num] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_num @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_510_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_511_order__less__subst2,axiom,
    ! [A2: real,B3: real,F: real > int,C: int] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_int @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_512_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_real @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_513_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_514_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_num @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_515_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_516_order__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_int @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_517_order__less__not__sym,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_518_order__less__not__sym,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_519_order__less__not__sym,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_520_order__less__not__sym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_521_order__less__not__sym,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_522_order__less__imp__triv,axiom,
    ! [X2: real,Y3: real,P: $o] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ( ord_less_real @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_523_order__less__imp__triv,axiom,
    ! [X2: rat,Y3: rat,P: $o] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_524_order__less__imp__triv,axiom,
    ! [X2: num,Y3: num,P: $o] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( ( ord_less_num @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_525_order__less__imp__triv,axiom,
    ! [X2: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_526_order__less__imp__triv,axiom,
    ! [X2: int,Y3: int,P: $o] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( ( ord_less_int @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_527_linorder__less__linear,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_real @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_528_linorder__less__linear,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_rat @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_529_linorder__less__linear,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_num @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_530_linorder__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_531_linorder__less__linear,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_int @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_532_order__less__imp__not__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_533_order__less__imp__not__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_534_order__less__imp__not__eq,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_535_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_536_order__less__imp__not__eq,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_537_order__less__imp__not__eq2,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_538_order__less__imp__not__eq2,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_539_order__less__imp__not__eq2,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_540_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_541_order__less__imp__not__eq2,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_542_order__less__imp__not__less,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_543_order__less__imp__not__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_544_order__less__imp__not__less,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_545_order__less__imp__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_546_order__less__imp__not__less,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_547_ex__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ? [X: set_nat] : ( member_set_nat @ X @ A3 ) )
      = ( A3 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_548_ex__in__conv,axiom,
    ! [A3: set_real] :
      ( ( ? [X: real] : ( member_real @ X @ A3 ) )
      = ( A3 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_549_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ? [X: $o] : ( member_o @ X @ A3 ) )
      = ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_550_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_551_ex__in__conv,axiom,
    ! [A3: set_int] :
      ( ( ? [X: int] : ( member_int @ X @ A3 ) )
      = ( A3 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_552_equals0I,axiom,
    ! [A3: set_set_nat] :
      ( ! [Y4: set_nat] :
          ~ ( member_set_nat @ Y4 @ A3 )
     => ( A3 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_553_equals0I,axiom,
    ! [A3: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A3 )
     => ( A3 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_554_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A3 )
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_555_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_556_equals0I,axiom,
    ! [A3: set_int] :
      ( ! [Y4: int] :
          ~ ( member_int @ Y4 @ A3 )
     => ( A3 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_557_equals0D,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( ( A3 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A2 @ A3 ) ) ).

% equals0D
thf(fact_558_equals0D,axiom,
    ! [A3: set_real,A2: real] :
      ( ( A3 = bot_bot_set_real )
     => ~ ( member_real @ A2 @ A3 ) ) ).

% equals0D
thf(fact_559_equals0D,axiom,
    ! [A3: set_o,A2: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A3 ) ) ).

% equals0D
thf(fact_560_equals0D,axiom,
    ! [A3: set_nat,A2: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A3 ) ) ).

% equals0D
thf(fact_561_equals0D,axiom,
    ! [A3: set_int,A2: int] :
      ( ( A3 = bot_bot_set_int )
     => ~ ( member_int @ A2 @ A3 ) ) ).

% equals0D
thf(fact_562_emptyE,axiom,
    ! [A2: set_nat] :
      ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_563_emptyE,axiom,
    ! [A2: real] :
      ~ ( member_real @ A2 @ bot_bot_set_real ) ).

% emptyE
thf(fact_564_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_565_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_566_emptyE,axiom,
    ! [A2: int] :
      ~ ( member_int @ A2 @ bot_bot_set_int ) ).

% emptyE
thf(fact_567_not__psubset__empty,axiom,
    ! [A3: set_real] :
      ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_568_not__psubset__empty,axiom,
    ! [A3: set_o] :
      ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_569_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_570_not__psubset__empty,axiom,
    ! [A3: set_int] :
      ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_571_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A5: set_nat] :
            ( ( finite_finite_nat @ A5 )
           => ( ! [B5: set_nat] :
                  ( ( ord_less_set_nat @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_572_finite__psubset__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ! [A5: set_int] :
            ( ( finite_finite_int @ A5 )
           => ( ! [B5: set_int] :
                  ( ( ord_less_set_int @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_573_finite__psubset__induct,axiom,
    ! [A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [A5: set_complex] :
            ( ( finite3207457112153483333omplex @ A5 )
           => ( ! [B5: set_complex] :
                  ( ( ord_less_set_complex @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_574_finite__psubset__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [A5: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A5 )
           => ( ! [B5: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_575_finite__psubset__induct,axiom,
    ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [A5: set_Extended_enat] :
            ( ( finite4001608067531595151d_enat @ A5 )
           => ( ! [B5: set_Extended_enat] :
                  ( ( ord_le2529575680413868914d_enat @ B5 @ A5 )
                 => ( P @ B5 ) )
             => ( P @ A5 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_576_leD,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ Y3 @ X2 )
     => ~ ( ord_less_real @ X2 @ Y3 ) ) ).

% leD
thf(fact_577_leD,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X2 )
     => ~ ( ord_less_set_int @ X2 @ Y3 ) ) ).

% leD
thf(fact_578_leD,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ~ ( ord_less_rat @ X2 @ Y3 ) ) ).

% leD
thf(fact_579_leD,axiom,
    ! [Y3: num,X2: num] :
      ( ( ord_less_eq_num @ Y3 @ X2 )
     => ~ ( ord_less_num @ X2 @ Y3 ) ) ).

% leD
thf(fact_580_leD,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).

% leD
thf(fact_581_leD,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ Y3 @ X2 )
     => ~ ( ord_less_int @ X2 @ Y3 ) ) ).

% leD
thf(fact_582_leI,axiom,
    ! [X2: real,Y3: real] :
      ( ~ ( ord_less_real @ X2 @ Y3 )
     => ( ord_less_eq_real @ Y3 @ X2 ) ) ).

% leI
thf(fact_583_leI,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X2 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).

% leI
thf(fact_584_leI,axiom,
    ! [X2: num,Y3: num] :
      ( ~ ( ord_less_num @ X2 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X2 ) ) ).

% leI
thf(fact_585_leI,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% leI
thf(fact_586_leI,axiom,
    ! [X2: int,Y3: int] :
      ( ~ ( ord_less_int @ X2 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X2 ) ) ).

% leI
thf(fact_587_nless__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ~ ( ord_less_real @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_588_nless__le,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ~ ( ord_less_set_int @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_589_nless__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ~ ( ord_less_rat @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_590_nless__le,axiom,
    ! [A2: num,B3: num] :
      ( ( ~ ( ord_less_num @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_591_nless__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_592_nless__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ~ ( ord_less_int @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B3 )
        | ( A2 = B3 ) ) ) ).

% nless_le
thf(fact_593_antisym__conv1,axiom,
    ! [X2: real,Y3: real] :
      ( ~ ( ord_less_real @ X2 @ Y3 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_594_antisym__conv1,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ~ ( ord_less_set_int @ X2 @ Y3 )
     => ( ( ord_less_eq_set_int @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_595_antisym__conv1,axiom,
    ! [X2: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X2 @ Y3 )
     => ( ( ord_less_eq_rat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_596_antisym__conv1,axiom,
    ! [X2: num,Y3: num] :
      ( ~ ( ord_less_num @ X2 @ Y3 )
     => ( ( ord_less_eq_num @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_597_antisym__conv1,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_598_antisym__conv1,axiom,
    ! [X2: int,Y3: int] :
      ( ~ ( ord_less_int @ X2 @ Y3 )
     => ( ( ord_less_eq_int @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_599_antisym__conv2,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ Y3 )
     => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_600_antisym__conv2,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ~ ( ord_less_set_int @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_601_antisym__conv2,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_602_antisym__conv2,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
     => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_603_antisym__conv2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_604_antisym__conv2,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_605_dense__ge,axiom,
    ! [Z: real,Y3: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z @ X5 )
         => ( ord_less_eq_real @ Y3 @ X5 ) )
     => ( ord_less_eq_real @ Y3 @ Z ) ) ).

% dense_ge
thf(fact_606_dense__ge,axiom,
    ! [Z: rat,Y3: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ Z @ X5 )
         => ( ord_less_eq_rat @ Y3 @ X5 ) )
     => ( ord_less_eq_rat @ Y3 @ Z ) ) ).

% dense_ge
thf(fact_607_dense__le,axiom,
    ! [Y3: real,Z: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y3 )
         => ( ord_less_eq_real @ X5 @ Z ) )
     => ( ord_less_eq_real @ Y3 @ Z ) ) ).

% dense_le
thf(fact_608_dense__le,axiom,
    ! [Y3: rat,Z: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Y3 )
         => ( ord_less_eq_rat @ X5 @ Z ) )
     => ( ord_less_eq_rat @ Y3 @ Z ) ) ).

% dense_le
thf(fact_609_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_610_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_611_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_612_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_613_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_614_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_615_not__le__imp__less,axiom,
    ! [Y3: real,X2: real] :
      ( ~ ( ord_less_eq_real @ Y3 @ X2 )
     => ( ord_less_real @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_616_not__le__imp__less,axiom,
    ! [Y3: rat,X2: rat] :
      ( ~ ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ord_less_rat @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_617_not__le__imp__less,axiom,
    ! [Y3: num,X2: num] :
      ( ~ ( ord_less_eq_num @ Y3 @ X2 )
     => ( ord_less_num @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_618_not__le__imp__less,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ord_less_nat @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_619_not__le__imp__less,axiom,
    ! [Y3: int,X2: int] :
      ( ~ ( ord_less_eq_int @ Y3 @ X2 )
     => ( ord_less_int @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_620_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_real @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_621_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_set_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_622_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_623_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_624_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_625_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_626_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_627_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_628_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_629_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_630_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_631_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_632_order_Ostrict__trans1,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_633_order_Ostrict__trans1,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_set_int @ B3 @ C )
       => ( ord_less_set_int @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_634_order_Ostrict__trans1,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_635_order_Ostrict__trans1,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_num @ B3 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_636_order_Ostrict__trans1,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_637_order_Ostrict__trans1,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_638_order_Ostrict__trans2,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_639_order_Ostrict__trans2,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C )
       => ( ord_less_set_int @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_640_order_Ostrict__trans2,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_641_order_Ostrict__trans2,axiom,
    ! [A2: num,B3: num,C: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_642_order_Ostrict__trans2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_643_order_Ostrict__trans2,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_644_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_645_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_646_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_647_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_648_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_649_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_650_dense__ge__bounded,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( ord_less_real @ Z @ X2 )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z @ W )
           => ( ( ord_less_real @ W @ X2 )
             => ( ord_less_eq_real @ Y3 @ W ) ) )
       => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_651_dense__ge__bounded,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( ord_less_rat @ Z @ X2 )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z @ W )
           => ( ( ord_less_rat @ W @ X2 )
             => ( ord_less_eq_rat @ Y3 @ W ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_652_dense__le__bounded,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ! [W: real] :
            ( ( ord_less_real @ X2 @ W )
           => ( ( ord_less_real @ W @ Y3 )
             => ( ord_less_eq_real @ W @ Z ) ) )
       => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_653_dense__le__bounded,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X2 @ W )
           => ( ( ord_less_rat @ W @ Y3 )
             => ( ord_less_eq_rat @ W @ Z ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_654_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_real @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_655_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( ord_less_set_int @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_656_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_rat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_657_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_num @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_658_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_659_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_int @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_660_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_661_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_662_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_663_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_664_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_665_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_666_dual__order_Ostrict__trans1,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ( ord_less_real @ C @ B3 )
       => ( ord_less_real @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_667_dual__order_Ostrict__trans1,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( ord_less_set_int @ C @ B3 )
       => ( ord_less_set_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_668_dual__order_Ostrict__trans1,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_rat @ C @ B3 )
       => ( ord_less_rat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_669_dual__order_Ostrict__trans1,axiom,
    ! [B3: num,A2: num,C: num] :
      ( ( ord_less_eq_num @ B3 @ A2 )
     => ( ( ord_less_num @ C @ B3 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_670_dual__order_Ostrict__trans1,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_nat @ C @ B3 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_671_dual__order_Ostrict__trans1,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_int @ C @ B3 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_672_dual__order_Ostrict__trans2,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_less_eq_real @ C @ B3 )
       => ( ord_less_real @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_673_dual__order_Ostrict__trans2,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( ( ord_less_set_int @ B3 @ A2 )
     => ( ( ord_less_eq_set_int @ C @ B3 )
       => ( ord_less_set_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_674_dual__order_Ostrict__trans2,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ C @ B3 )
       => ( ord_less_rat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_675_dual__order_Ostrict__trans2,axiom,
    ! [B3: num,A2: num,C: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ( ( ord_less_eq_num @ C @ B3 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_676_dual__order_Ostrict__trans2,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_677_dual__order_Ostrict__trans2,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ C @ B3 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_678_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_679_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A4 )
          & ~ ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_680_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_681_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_682_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_683_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_684_order_Ostrict__implies__order,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_eq_real @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_685_order_Ostrict__implies__order,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_686_order_Ostrict__implies__order,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_687_order_Ostrict__implies__order,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ord_less_eq_num @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_688_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_689_order_Ostrict__implies__order,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ord_less_eq_int @ A2 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_690_dual__order_Ostrict__implies__order,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ord_less_eq_real @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_691_dual__order_Ostrict__implies__order,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( ord_less_set_int @ B3 @ A2 )
     => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_692_dual__order_Ostrict__implies__order,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ord_less_eq_rat @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_693_dual__order_Ostrict__implies__order,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ( ord_less_eq_num @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_694_dual__order_Ostrict__implies__order,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_695_dual__order_Ostrict__implies__order,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ord_less_eq_int @ B3 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_696_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_697_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_set_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_698_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_699_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_num @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_700_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_701_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_702_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_703_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_704_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_705_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_706_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_707_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_708_linorder__not__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
      = ( ord_less_real @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_709_linorder__not__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ~ ( ord_less_eq_rat @ X2 @ Y3 ) )
      = ( ord_less_rat @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_710_linorder__not__le,axiom,
    ! [X2: num,Y3: num] :
      ( ( ~ ( ord_less_eq_num @ X2 @ Y3 ) )
      = ( ord_less_num @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_711_linorder__not__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_712_linorder__not__le,axiom,
    ! [X2: int,Y3: int] :
      ( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
      = ( ord_less_int @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_713_linorder__not__less,axiom,
    ! [X2: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
      = ( ord_less_eq_real @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_714_linorder__not__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
      = ( ord_less_eq_rat @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_715_linorder__not__less,axiom,
    ! [X2: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
      = ( ord_less_eq_num @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_716_linorder__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_717_linorder__not__less,axiom,
    ! [X2: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
      = ( ord_less_eq_int @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_718_order__less__imp__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_719_order__less__imp__le,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_set_int @ X2 @ Y3 )
     => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_720_order__less__imp__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_721_order__less__imp__le,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( ord_less_eq_num @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_722_order__less__imp__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_723_order__less__imp__le,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( ord_less_eq_int @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_724_order__le__neq__trans,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_real @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_725_order__le__neq__trans,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_set_int @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_726_order__le__neq__trans,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_rat @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_727_order__le__neq__trans,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_num @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_728_order__le__neq__trans,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_729_order__le__neq__trans,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_int @ A2 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_730_order__neq__le__trans,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_real @ A2 @ B3 )
       => ( ord_less_real @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_731_order__neq__le__trans,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ord_less_set_int @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_732_order__neq__le__trans,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_rat @ A2 @ B3 )
       => ( ord_less_rat @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_733_order__neq__le__trans,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_num @ A2 @ B3 )
       => ( ord_less_num @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_734_order__neq__le__trans,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_735_order__neq__le__trans,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 != B3 )
     => ( ( ord_less_eq_int @ A2 @ B3 )
       => ( ord_less_int @ A2 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_736_order__le__less__trans,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( ord_less_eq_real @ X2 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z )
       => ( ord_less_real @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_737_order__le__less__trans,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ord_less_set_int @ Y3 @ Z )
       => ( ord_less_set_int @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_738_order__le__less__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z )
       => ( ord_less_rat @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_739_order__le__less__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z )
       => ( ord_less_num @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_740_order__le__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_741_order__le__less__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ X2 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_742_order__less__le__trans,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ Z )
       => ( ord_less_real @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_743_order__less__le__trans,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X2 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ Z )
       => ( ord_less_set_int @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_744_order__less__le__trans,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z )
       => ( ord_less_rat @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_745_order__less__le__trans,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( ord_less_num @ X2 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z )
       => ( ord_less_num @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_746_order__less__le__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_747_order__less__le__trans,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_int @ X2 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_int @ X2 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_748_order__le__less__subst1,axiom,
    ! [A2: real,F: real > real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_749_order__le__less__subst1,axiom,
    ! [A2: real,F: rat > real,B3: rat,C: rat] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_750_order__le__less__subst1,axiom,
    ! [A2: real,F: num > real,B3: num,C: num] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_751_order__le__less__subst1,axiom,
    ! [A2: real,F: nat > real,B3: nat,C: nat] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_752_order__le__less__subst1,axiom,
    ! [A2: real,F: int > real,B3: int,C: int] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_int @ B3 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_753_order__le__less__subst1,axiom,
    ! [A2: rat,F: real > rat,B3: real,C: real] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_real @ B3 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_754_order__le__less__subst1,axiom,
    ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_755_order__le__less__subst1,axiom,
    ! [A2: rat,F: num > rat,B3: num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_756_order__le__less__subst1,axiom,
    ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_757_order__le__less__subst1,axiom,
    ! [A2: rat,F: int > rat,B3: int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_int @ B3 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_758_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_real @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_759_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_760_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_num @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_761_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_762_order__le__less__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_int @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_763_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_real @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_764_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_765_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_num @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_766_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_767_order__le__less__subst2,axiom,
    ! [A2: num,B3: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_less_int @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_768_order__less__le__subst1,axiom,
    ! [A2: real,F: rat > real,B3: rat,C: rat] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_769_order__less__le__subst1,axiom,
    ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_770_order__less__le__subst1,axiom,
    ! [A2: num,F: rat > num,B3: rat,C: rat] :
      ( ( ord_less_num @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_771_order__less__le__subst1,axiom,
    ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
      ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_772_order__less__le__subst1,axiom,
    ! [A2: int,F: rat > int,B3: rat,C: rat] :
      ( ( ord_less_int @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_773_order__less__le__subst1,axiom,
    ! [A2: real,F: num > real,B3: num,C: num] :
      ( ( ord_less_real @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_774_order__less__le__subst1,axiom,
    ! [A2: rat,F: num > rat,B3: num,C: num] :
      ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_775_order__less__le__subst1,axiom,
    ! [A2: num,F: num > num,B3: num,C: num] :
      ( ( ord_less_num @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_776_order__less__le__subst1,axiom,
    ! [A2: nat,F: num > nat,B3: num,C: num] :
      ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_777_order__less__le__subst1,axiom,
    ! [A2: int,F: num > int,B3: num,C: num] :
      ( ( ord_less_int @ A2 @ ( F @ B3 ) )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_778_order__less__le__subst2,axiom,
    ! [A2: real,B3: real,F: real > real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_779_order__less__le__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_780_order__less__le__subst2,axiom,
    ! [A2: num,B3: num,F: num > real,C: real] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_781_order__less__le__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_782_order__less__le__subst2,axiom,
    ! [A2: int,B3: int,F: int > real,C: real] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_783_order__less__le__subst2,axiom,
    ! [A2: real,B3: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_784_order__less__le__subst2,axiom,
    ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_785_order__less__le__subst2,axiom,
    ! [A2: num,B3: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_786_order__less__le__subst2,axiom,
    ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_787_order__less__le__subst2,axiom,
    ! [A2: int,B3: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_788_linorder__le__less__linear,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ Y3 )
      | ( ord_less_real @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_789_linorder__le__less__linear,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
      | ( ord_less_rat @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_790_linorder__le__less__linear,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
      | ( ord_less_num @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_791_linorder__le__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_792_linorder__le__less__linear,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
      | ( ord_less_int @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_793_order__le__imp__less__or__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ Y3 )
     => ( ( ord_less_real @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_794_order__le__imp__less__or__eq,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ord_less_set_int @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_795_order__le__imp__less__or__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ord_less_rat @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_796_order__le__imp__less__or__eq,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
     => ( ( ord_less_num @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_797_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_798_order__le__imp__less__or__eq,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ord_less_int @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_799_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
     => ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_800_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_801_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_802_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_803_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_804_bot_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_805_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_806_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_807_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_808_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_809_bot_Oextremum,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% bot.extremum
thf(fact_810_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_811_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_812_bot_Oextremum,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% bot.extremum
thf(fact_813_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_814_bot_Oextremum__strict,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_815_bot_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_816_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_817_bot_Oextremum__strict,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_818_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_819_bot_Onot__eq__extremum,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_820_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_821_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_822_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_823_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_824_finite__has__minimal2,axiom,
    ! [A3: set_real,A2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A2 @ A3 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A3 )
            & ( ord_less_eq_real @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_825_finite__has__minimal2,axiom,
    ! [A3: set_o,A2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ A2 @ A3 )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A3 )
            & ( ord_less_eq_o @ X5 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A3 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_826_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A2 @ A3 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A3 )
            & ( ord_less_eq_set_nat @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_827_finite__has__minimal2,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ A2 @ A3 )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
            & ( ord_le2932123472753598470d_enat @ X5 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_828_finite__has__minimal2,axiom,
    ! [A3: set_set_int,A2: set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( member_set_int @ A2 @ A3 )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
            & ( ord_less_eq_set_int @ X5 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A3 )
               => ( ( ord_less_eq_set_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_829_finite__has__minimal2,axiom,
    ! [A3: set_rat,A2: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A2 @ A3 )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A3 )
            & ( ord_less_eq_rat @ X5 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_830_finite__has__minimal2,axiom,
    ! [A3: set_num,A2: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A2 @ A3 )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A3 )
            & ( ord_less_eq_num @ X5 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_831_finite__has__minimal2,axiom,
    ! [A3: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A2 @ A3 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ( ord_less_eq_nat @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_832_finite__has__minimal2,axiom,
    ! [A3: set_int,A2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A2 @ A3 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A3 )
            & ( ord_less_eq_int @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_833_finite__has__maximal2,axiom,
    ! [A3: set_real,A2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ A2 @ A3 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A3 )
            & ( ord_less_eq_real @ A2 @ X5 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A3 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_834_finite__has__maximal2,axiom,
    ! [A3: set_o,A2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ A2 @ A3 )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A3 )
            & ( ord_less_eq_o @ A2 @ X5 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A3 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_835_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A2 @ A3 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A3 )
            & ( ord_less_eq_set_nat @ A2 @ X5 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_836_finite__has__maximal2,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ A2 @ A3 )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
            & ( ord_le2932123472753598470d_enat @ A2 @ X5 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A3 )
               => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_837_finite__has__maximal2,axiom,
    ! [A3: set_set_int,A2: set_int] :
      ( ( finite6197958912794628473et_int @ A3 )
     => ( ( member_set_int @ A2 @ A3 )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
            & ( ord_less_eq_set_int @ A2 @ X5 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A3 )
               => ( ( ord_less_eq_set_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_838_finite__has__maximal2,axiom,
    ! [A3: set_rat,A2: rat] :
      ( ( finite_finite_rat @ A3 )
     => ( ( member_rat @ A2 @ A3 )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A3 )
            & ( ord_less_eq_rat @ A2 @ X5 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A3 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_839_finite__has__maximal2,axiom,
    ! [A3: set_num,A2: num] :
      ( ( finite_finite_num @ A3 )
     => ( ( member_num @ A2 @ A3 )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A3 )
            & ( ord_less_eq_num @ A2 @ X5 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A3 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_840_finite__has__maximal2,axiom,
    ! [A3: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A2 @ A3 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ( ord_less_eq_nat @ A2 @ X5 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_841_finite__has__maximal2,axiom,
    ! [A3: set_int,A2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ A2 @ A3 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A3 )
            & ( ord_less_eq_int @ A2 @ X5 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A3 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_842_infinite__imp__nonempty,axiom,
    ! [S: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S )
     => ( S != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_843_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_844_infinite__imp__nonempty,axiom,
    ! [S: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S )
     => ( S != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
thf(fact_845_infinite__imp__nonempty,axiom,
    ! [S: set_real] :
      ( ~ ( finite_finite_real @ S )
     => ( S != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_846_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ ( finite_finite_o @ S )
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_847_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_848_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_849_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_850_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_851_finite_OemptyI,axiom,
    finite4001608067531595151d_enat @ bot_bo7653980558646680370d_enat ).

% finite.emptyI
thf(fact_852_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_853_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_854_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_855_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_856_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X2 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
          | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_857_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).

% buildup_nothing_in_min_max
thf(fact_858_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X2 ) @ Y3 )
        = ( ( X2 != Y3 )
          & ( vEBT_vebt_member @ T @ Y3 ) ) ) ) ).

% dele_member_cont_corr
thf(fact_859_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M2: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N4 )
         => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_860_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M2: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_861_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N5: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
              & ( member_nat @ N5 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_862_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N6 )
         => ( ord_less_nat @ X5 @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_863_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M2: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_864_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M2: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N4 )
         => ( ord_less_nat @ X @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_865_arg__min__if__finite_I2_J,axiom,
    ! [S: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( S != bot_bot_set_complex )
       => ~ ? [X4: complex] :
              ( ( member_complex @ X4 @ S )
              & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_866_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ~ ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ S )
              & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic1189837152898106425t_real @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_867_arg__min__if__finite_I2_J,axiom,
    ! [S: set_real,F: real > real] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ~ ? [X4: real] :
              ( ( member_real @ X4 @ S )
              & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_868_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > real] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( ( member_o @ X4 @ S )
              & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8697145971487455083o_real @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_869_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ S )
              & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_870_arg__min__if__finite_I2_J,axiom,
    ! [S: set_int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ~ ? [X4: int] :
              ( ( member_int @ X4 @ S )
              & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_871_arg__min__if__finite_I2_J,axiom,
    ! [S: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( S != bot_bot_set_complex )
       => ~ ? [X4: complex] :
              ( ( member_complex @ X4 @ S )
              & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_872_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ~ ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ S )
              & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic3210252021154270693at_rat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_873_arg__min__if__finite_I2_J,axiom,
    ! [S: set_real,F: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ~ ? [X4: real] :
              ( ( member_real @ X4 @ S )
              & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_874_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X4: $o] :
              ( ( member_o @ X4 @ S )
              & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_875_arg__min__least,axiom,
    ! [S: set_complex,Y3: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( S != bot_bot_set_complex )
       => ( ( member_complex @ Y3 @ S )
         => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_876_arg__min__least,axiom,
    ! [S: set_Extended_enat,Y3: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y3 @ S )
         => ( ord_less_eq_rat @ ( F @ ( lattic3210252021154270693at_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_877_arg__min__least,axiom,
    ! [S: set_real,Y3: real,F: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ( ( member_real @ Y3 @ S )
         => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_878_arg__min__least,axiom,
    ! [S: set_o,Y3: $o,F: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y3 @ S )
         => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_879_arg__min__least,axiom,
    ! [S: set_nat,Y3: nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y3 @ S )
         => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_880_arg__min__least,axiom,
    ! [S: set_int,Y3: int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ( ( member_int @ Y3 @ S )
         => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_881_arg__min__least,axiom,
    ! [S: set_complex,Y3: complex,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( S != bot_bot_set_complex )
       => ( ( member_complex @ Y3 @ S )
         => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_882_arg__min__least,axiom,
    ! [S: set_Extended_enat,Y3: extended_enat,F: extended_enat > num] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( S != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y3 @ S )
         => ( ord_less_eq_num @ ( F @ ( lattic402713867396545063at_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_883_arg__min__least,axiom,
    ! [S: set_real,Y3: real,F: real > num] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ( ( member_real @ Y3 @ S )
         => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_884_arg__min__least,axiom,
    ! [S: set_o,Y3: $o,F: $o > num] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y3 @ S )
         => ( ord_less_eq_num @ ( F @ ( lattic8556559851467007577_o_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_885_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K2 @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_886_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X2 ) @ N ) ) ).

% delete_pres_valid
thf(fact_887_subsetI,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( member_real @ X5 @ B2 ) )
     => ( ord_less_eq_set_real @ A3 @ B2 ) ) ).

% subsetI
thf(fact_888_subsetI,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( member_o @ X5 @ B2 ) )
     => ( ord_less_eq_set_o @ A3 @ B2 ) ) ).

% subsetI
thf(fact_889_subsetI,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A3 )
         => ( member_set_nat @ X5 @ B2 ) )
     => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ).

% subsetI
thf(fact_890_subsetI,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ( member_nat @ X5 @ B2 ) )
     => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).

% subsetI
thf(fact_891_subsetI,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( member_int @ X5 @ B2 ) )
     => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).

% subsetI
thf(fact_892_subset__antisym,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% subset_antisym
thf(fact_893_psubsetI,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( A3 != B2 )
       => ( ord_less_set_int @ A3 @ B2 ) ) ) ).

% psubsetI
thf(fact_894_in__mono,axiom,
    ! [A3: set_real,B2: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A3 @ B2 )
     => ( ( member_real @ X2 @ A3 )
       => ( member_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_895_in__mono,axiom,
    ! [A3: set_o,B2: set_o,X2: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B2 )
     => ( ( member_o @ X2 @ A3 )
       => ( member_o @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_896_in__mono,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,X2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
     => ( ( member_set_nat @ X2 @ A3 )
       => ( member_set_nat @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_897_in__mono,axiom,
    ! [A3: set_nat,B2: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( member_nat @ X2 @ A3 )
       => ( member_nat @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_898_in__mono,axiom,
    ! [A3: set_int,B2: set_int,X2: int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( member_int @ X2 @ A3 )
       => ( member_int @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_899_subsetD,axiom,
    ! [A3: set_real,B2: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A3 @ B2 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_900_subsetD,axiom,
    ! [A3: set_o,B2: set_o,C: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B2 )
     => ( ( member_o @ C @ A3 )
       => ( member_o @ C @ B2 ) ) ) ).

% subsetD
thf(fact_901_subsetD,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_902_subsetD,axiom,
    ! [A3: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_903_subsetD,axiom,
    ! [A3: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B2 ) ) ) ).

% subsetD
thf(fact_904_psubsetD,axiom,
    ! [A3: set_real,B2: set_real,C: real] :
      ( ( ord_less_set_real @ A3 @ B2 )
     => ( ( member_real @ C @ A3 )
       => ( member_real @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_905_psubsetD,axiom,
    ! [A3: set_o,B2: set_o,C: $o] :
      ( ( ord_less_set_o @ A3 @ B2 )
     => ( ( member_o @ C @ A3 )
       => ( member_o @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_906_psubsetD,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B2 )
     => ( ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_907_psubsetD,axiom,
    ! [A3: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A3 @ B2 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_908_psubsetD,axiom,
    ! [A3: set_int,B2: set_int,C: int] :
      ( ( ord_less_set_int @ A3 @ B2 )
     => ( ( member_int @ C @ A3 )
       => ( member_int @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_909_psubsetE,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A3 @ B2 )
     => ~ ( ( ord_less_eq_set_int @ A3 @ B2 )
         => ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ).

% psubsetE
thf(fact_910_equalityE,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
     => ~ ( ( ord_less_eq_set_int @ A3 @ B2 )
         => ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ).

% equalityE
thf(fact_911_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A6 )
         => ( member_real @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_912_subset__eq,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [X: $o] :
          ( ( member_o @ X @ A6 )
         => ( member_o @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_913_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [X: set_nat] :
          ( ( member_set_nat @ X @ A6 )
         => ( member_set_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_914_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A6 )
         => ( member_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_915_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A6 )
         => ( member_int @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_916_equalityD1,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
     => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).

% equalityD1
thf(fact_917_equalityD2,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( A3 = B2 )
     => ( ord_less_eq_set_int @ B2 @ A3 ) ) ).

% equalityD2
thf(fact_918_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_919_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T3: real] :
          ( ( member_real @ T3 @ A6 )
         => ( member_real @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_920_subset__iff,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
        ! [T3: $o] :
          ( ( member_o @ T3 @ A6 )
         => ( member_o @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_921_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [T3: set_nat] :
          ( ( member_set_nat @ T3 @ A6 )
         => ( member_set_nat @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_922_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T3: nat] :
          ( ( member_nat @ T3 @ A6 )
         => ( member_nat @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_923_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T3: int] :
          ( ( member_int @ T3 @ A6 )
         => ( member_int @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_924_subset__refl,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).

% subset_refl
thf(fact_925_Collect__mono,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_926_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_927_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_928_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_929_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_930_subset__trans,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).

% subset_trans
thf(fact_931_set__eq__subset,axiom,
    ( ( ^ [Y6: set_int,Z3: set_int] : Y6 = Z3 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_932_Collect__mono__iff,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
      = ( ! [X: real] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_933_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_934_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X: set_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_935_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_936_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_937_psubset__imp__subset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A3 @ B2 )
     => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).

% psubset_imp_subset
thf(fact_938_psubset__subset__trans,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_set_int @ A3 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ord_less_set_int @ A3 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_939_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_940_subset__psubset__trans,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( ord_less_set_int @ B2 @ C2 )
       => ( ord_less_set_int @ A3 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_941_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_942_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M5: nat] :
      ( ( P @ X2 )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M5 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_943_finite__transitivity__chain,axiom,
    ! [A3: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ! [X5: set_nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A3 )
               => ? [Y5: set_nat] :
                    ( ( member_set_nat @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_944_finite__transitivity__chain,axiom,
    ! [A3: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X5: complex] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: complex,Y4: complex,Z4: complex] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ A3 )
               => ? [Y5: complex] :
                    ( ( member_complex @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_945_finite__transitivity__chain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [X5: product_prod_nat_nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat,Z4: product_prod_nat_nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ A3 )
               => ? [Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_946_finite__transitivity__chain,axiom,
    ! [A3: set_Extended_enat,R: extended_enat > extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X5: extended_enat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: extended_enat,Y4: extended_enat,Z4: extended_enat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ A3 )
               => ? [Y5: extended_enat] :
                    ( ( member_Extended_enat @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_947_finite__transitivity__chain,axiom,
    ! [A3: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X5: real] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: real,Y4: real,Z4: real] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ A3 )
               => ? [Y5: real] :
                    ( ( member_real @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_948_finite__transitivity__chain,axiom,
    ! [A3: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X5: $o] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: $o,Y4: $o,Z4: $o] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ A3 )
               => ? [Y5: $o] :
                    ( ( member_o @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_949_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X5: nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z4: nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ A3 )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_950_finite__transitivity__chain,axiom,
    ! [A3: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X5: int] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: int,Y4: int,Z4: int] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ A3 )
               => ? [Y5: int] :
                    ( ( member_int @ Y5 @ A3 )
                    & ( R @ X5 @ Y5 ) ) )
           => ( A3 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_951_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_952_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_953_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_954_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_955_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_956_Collect__empty__eq__bot,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( P = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_957_Collect__empty__eq__bot,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( P = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_958_Collect__empty__eq__bot,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( P = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_959_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_960_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_961_Collect__empty__eq__bot,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( P = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_962_subset__emptyI,axiom,
    ! [A3: set_set_nat] :
      ( ! [X5: set_nat] :
          ~ ( member_set_nat @ X5 @ A3 )
     => ( ord_le6893508408891458716et_nat @ A3 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_963_subset__emptyI,axiom,
    ! [A3: set_real] :
      ( ! [X5: real] :
          ~ ( member_real @ X5 @ A3 )
     => ( ord_less_eq_set_real @ A3 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_964_subset__emptyI,axiom,
    ! [A3: set_o] :
      ( ! [X5: $o] :
          ~ ( member_o @ X5 @ A3 )
     => ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_965_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X5: nat] :
          ~ ( member_nat @ X5 @ A3 )
     => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_966_subset__emptyI,axiom,
    ! [A3: set_int] :
      ( ! [X5: int] :
          ~ ( member_int @ X5 @ A3 )
     => ( ord_less_eq_set_int @ A3 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_967_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ? [E: real] :
            ( ( ord_less_real @ zero_zero_real @ E )
            & ( ord_less_real @ E @ D1 )
            & ( ord_less_real @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_968_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D2 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_969_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_970_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_971_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_972_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_973_complete__interval,axiom,
    ! [A2: real,B3: real,P: real > $o] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B3 )
         => ? [C3: real] :
              ( ( ord_less_eq_real @ A2 @ C3 )
              & ( ord_less_eq_real @ C3 @ B3 )
              & ! [X4: real] :
                  ( ( ( ord_less_eq_real @ A2 @ X4 )
                    & ( ord_less_real @ X4 @ C3 ) )
                 => ( P @ X4 ) )
              & ! [D3: real] :
                  ( ! [X5: real] :
                      ( ( ( ord_less_eq_real @ A2 @ X5 )
                        & ( ord_less_real @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_974_complete__interval,axiom,
    ! [A2: nat,B3: nat,P: nat > $o] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B3 )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A2 @ C3 )
              & ( ord_less_eq_nat @ C3 @ B3 )
              & ! [X4: nat] :
                  ( ( ( ord_less_eq_nat @ A2 @ X4 )
                    & ( ord_less_nat @ X4 @ C3 ) )
                 => ( P @ X4 ) )
              & ! [D3: nat] :
                  ( ! [X5: nat] :
                      ( ( ( ord_less_eq_nat @ A2 @ X5 )
                        & ( ord_less_nat @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_975_complete__interval,axiom,
    ! [A2: int,B3: int,P: int > $o] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( P @ A2 )
       => ( ~ ( P @ B3 )
         => ? [C3: int] :
              ( ( ord_less_eq_int @ A2 @ C3 )
              & ( ord_less_eq_int @ C3 @ B3 )
              & ! [X4: int] :
                  ( ( ( ord_less_eq_int @ A2 @ X4 )
                    & ( ord_less_int @ X4 @ C3 ) )
                 => ( P @ X4 ) )
              & ! [D3: int] :
                  ( ! [X5: int] :
                      ( ( ( ord_less_eq_int @ A2 @ X5 )
                        & ( ord_less_int @ X5 @ D3 ) )
                     => ( P @ X5 ) )
                 => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_976_verit__comp__simplify1_I3_J,axiom,
    ! [B7: real,A7: real] :
      ( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
      = ( ord_less_real @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_977_verit__comp__simplify1_I3_J,axiom,
    ! [B7: rat,A7: rat] :
      ( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
      = ( ord_less_rat @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_978_verit__comp__simplify1_I3_J,axiom,
    ! [B7: num,A7: num] :
      ( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
      = ( ord_less_num @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_979_verit__comp__simplify1_I3_J,axiom,
    ! [B7: nat,A7: nat] :
      ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
      = ( ord_less_nat @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_980_verit__comp__simplify1_I3_J,axiom,
    ! [B7: int,A7: int] :
      ( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
      = ( ord_less_int @ A7 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_981_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ~ ( ord_less_eq_real @ X4 @ T ) ) ).

% pinf(6)
thf(fact_982_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_983_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ~ ( ord_less_eq_num @ X4 @ T ) ) ).

% pinf(6)
thf(fact_984_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_985_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T ) ) ).

% pinf(6)
thf(fact_986_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ord_less_eq_real @ T @ X4 ) ) ).

% pinf(8)
thf(fact_987_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ord_less_eq_rat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_988_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( ord_less_eq_num @ T @ X4 ) ) ).

% pinf(8)
thf(fact_989_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_990_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ord_less_eq_int @ T @ X4 ) ) ).

% pinf(8)
thf(fact_991_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ord_less_eq_real @ X4 @ T ) ) ).

% minf(6)
thf(fact_992_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ord_less_eq_rat @ X4 @ T ) ) ).

% minf(6)
thf(fact_993_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( ord_less_eq_num @ X4 @ T ) ) ).

% minf(6)
thf(fact_994_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_995_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ord_less_eq_int @ X4 @ T ) ) ).

% minf(6)
thf(fact_996_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ~ ( ord_less_eq_real @ T @ X4 ) ) ).

% minf(8)
thf(fact_997_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).

% minf(8)
thf(fact_998_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ~ ( ord_less_eq_num @ T @ X4 ) ) ).

% minf(8)
thf(fact_999_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_1000_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ~ ( ord_less_eq_int @ T @ X4 ) ) ).

% minf(8)
thf(fact_1001_verit__la__disequality,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 = B3 )
      | ~ ( ord_less_eq_rat @ A2 @ B3 )
      | ~ ( ord_less_eq_rat @ B3 @ A2 ) ) ).

% verit_la_disequality
thf(fact_1002_verit__la__disequality,axiom,
    ! [A2: num,B3: num] :
      ( ( A2 = B3 )
      | ~ ( ord_less_eq_num @ A2 @ B3 )
      | ~ ( ord_less_eq_num @ B3 @ A2 ) ) ).

% verit_la_disequality
thf(fact_1003_verit__la__disequality,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
      | ~ ( ord_less_eq_nat @ A2 @ B3 )
      | ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% verit_la_disequality
thf(fact_1004_verit__la__disequality,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
      | ~ ( ord_less_eq_int @ A2 @ B3 )
      | ~ ( ord_less_eq_int @ B3 @ A2 ) ) ).

% verit_la_disequality
thf(fact_1005_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1006_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1007_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1008_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1009_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_1010_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ~ ( ord_less_real @ T @ X4 ) ) ).

% minf(7)
thf(fact_1011_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ~ ( ord_less_rat @ T @ X4 ) ) ).

% minf(7)
thf(fact_1012_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ~ ( ord_less_num @ T @ X4 ) ) ).

% minf(7)
thf(fact_1013_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ~ ( ord_less_nat @ T @ X4 ) ) ).

% minf(7)
thf(fact_1014_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ~ ( ord_less_int @ T @ X4 ) ) ).

% minf(7)
thf(fact_1015_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ord_less_real @ X4 @ T ) ) ).

% minf(5)
thf(fact_1016_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ord_less_rat @ X4 @ T ) ) ).

% minf(5)
thf(fact_1017_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( ord_less_num @ X4 @ T ) ) ).

% minf(5)
thf(fact_1018_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ord_less_nat @ X4 @ T ) ) ).

% minf(5)
thf(fact_1019_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ord_less_int @ X4 @ T ) ) ).

% minf(5)
thf(fact_1020_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1021_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1022_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1023_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1024_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1025_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1026_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1027_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1028_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1029_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1030_minf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1031_minf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1032_minf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1033_minf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1034_minf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1035_minf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1036_minf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1037_minf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1038_minf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1039_minf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1040_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ord_less_real @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1041_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ord_less_rat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1042_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( ord_less_num @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1043_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ord_less_nat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1044_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ord_less_int @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1045_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ~ ( ord_less_real @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1046_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ~ ( ord_less_rat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1047_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ~ ( ord_less_num @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1048_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1049_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ~ ( ord_less_int @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1050_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1051_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1052_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1053_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1054_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1055_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1056_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1057_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1058_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1059_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1060_pinf_I2_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1061_pinf_I2_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1062_pinf_I2_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1063_pinf_I2_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1064_pinf_I2_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1065_pinf_I1_J,axiom,
    ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1066_pinf_I1_J,axiom,
    ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1067_pinf_I1_J,axiom,
    ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1068_pinf_I1_J,axiom,
    ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1069_pinf_I1_J,axiom,
    ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P4 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q2 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P4 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1070_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_1071_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_1072_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_1073_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_1074_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_1075_ex__gt__or__lt,axiom,
    ! [A2: real] :
    ? [B: real] :
      ( ( ord_less_real @ A2 @ B )
      | ( ord_less_real @ B @ A2 ) ) ).

% ex_gt_or_lt
thf(fact_1076_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1077_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1078_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1079_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1080_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T3: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T3 @ X )
          | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_1081_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
       => ( vEBT_vebt_member @ T @ X2 ) ) ) ).

% valid_member_both_member_options
thf(fact_1082_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
        = ( vEBT_vebt_member @ T @ X2 ) ) ) ).

% both_member_options_equiv_member
thf(fact_1083_Leaf__0__not,axiom,
    ! [A2: $o,B3: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B3 ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1084_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X2 ) @ Y3 )
        = ( ( X2 != Y3 )
          & ( vEBT_V8194947554948674370ptions @ T @ Y3 ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_1085_maxbmo,axiom,
    ! [T: vEBT_VEBT,X2: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X2 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).

% maxbmo
thf(fact_1086_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1087_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1088_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1089_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
        = Y3 )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y3 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y3 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A ) )
               => ! [B: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B ) )
                   => ( Y3
                     != ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1090_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
        = Y3 )
     => ( ( ( Xa2 = none_num )
         => ( Y3 != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y3 != none_num ) ) )
         => ~ ! [A: num] :
                ( ( Xa2
                  = ( some_num @ A ) )
               => ! [B: num] :
                    ( ( Xb
                      = ( some_num @ B ) )
                   => ( Y3
                     != ( some_num @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1091_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
        = Y3 )
     => ( ( ( Xa2 = none_nat )
         => ( Y3 != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y3 != none_nat ) ) )
         => ~ ! [A: nat] :
                ( ( Xa2
                  = ( some_nat @ A ) )
               => ! [B: nat] :
                    ( ( Xb
                      = ( some_nat @ B ) )
                   => ( Y3
                     != ( some_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1092_Set_Ois__empty__def,axiom,
    ( is_empty_real
    = ( ^ [A6: set_real] : A6 = bot_bot_set_real ) ) ).

% Set.is_empty_def
thf(fact_1093_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A6: set_o] : A6 = bot_bot_set_o ) ) ).

% Set.is_empty_def
thf(fact_1094_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A6: set_nat] : A6 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
thf(fact_1095_Set_Ois__empty__def,axiom,
    ( is_empty_int
    = ( ^ [A6: set_int] : A6 = bot_bot_set_int ) ) ).

% Set.is_empty_def
thf(fact_1096_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_1097_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_1098_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_1099_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_1100_enumerate__mono__iff,axiom,
    ! [S: set_Extended_enat,M: nat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ S )
     => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% enumerate_mono_iff
thf(fact_1101_enumerate__mono__iff,axiom,
    ! [S: set_nat,M: nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% enumerate_mono_iff
thf(fact_1102_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_1103_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_1104_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_1105_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_1106_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_1107_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_1108_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_1109_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_1110_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_1111_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_1112_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_1113_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1114_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1115_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1116_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1117_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_1118_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_1119_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_1120_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_1121_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1122_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1123_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1124_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1125_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1126_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1127_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1128_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1129_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_1130_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_1131_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_1132_enumerate__in__set,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ S )
     => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ S ) ) ).

% enumerate_in_set
thf(fact_1133_enumerate__in__set,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).

% enumerate_in_set
thf(fact_1134_enumerate__Ex,axiom,
    ! [S: set_nat,S2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( member_nat @ S2 @ S )
       => ? [N3: nat] :
            ( ( infini8530281810654367211te_nat @ S @ N3 )
            = S2 ) ) ) ).

% enumerate_Ex
thf(fact_1135_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_1136_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_1137_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_1138_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_1139_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_1140_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_1141_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_1142_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_1143_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1144_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1145_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1146_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_1147_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1148_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1149_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1150_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_1151_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_1152_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_1153_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_1154_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_1155_le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).

% le_enumerate
thf(fact_1156_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A2 ) @ ( some_P7363390416028606310at_nat @ B3 ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A2 @ B3 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1157_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A2: num,B3: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A2 ) @ ( some_num @ B3 ) )
      = ( some_num @ ( F @ A2 @ B3 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1158_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A2: nat,B3: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A2 ) @ ( some_nat @ B3 ) )
      = ( some_nat @ ( F @ A2 @ B3 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1159_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1160_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: num > num > num,Uv: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1161_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1162_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_1163_enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_Extended_enat] :
      ( ( ord_less_nat @ M @ N )
     => ( ~ ( finite4001608067531595151d_enat @ S )
       => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ).

% enumerate_mono
thf(fact_1164_enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ~ ( finite_finite_nat @ S )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).

% enumerate_mono
thf(fact_1165_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_1166_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B3: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B3 ) ) ).

% vebt_delete.simps(1)
thf(fact_1167_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_1168_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1169_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1170_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1171_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A4: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_1172_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A: $o,B: $o] :
          ( T
          = ( vEBT_Leaf @ A @ B ) ) ) ).

% deg_1_Leaf
thf(fact_1173_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A: $o,B: $o] :
            ( T
            = ( vEBT_Leaf @ A @ B ) ) ) ) ).

% deg_1_Leafy
thf(fact_1174_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% pos_int_cases
thf(fact_1175_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( K
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_1176_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_1177_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_1178_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_1179_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_1180_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_1181_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1182_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1183_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_1184_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_1185_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_1186_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1187_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1188_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_1189_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_1190_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_1191_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1192_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nonneg_int_cases
thf(fact_1193_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_1194_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_1195_one__reorient,axiom,
    ! [X2: complex] :
      ( ( one_one_complex = X2 )
      = ( X2 = one_one_complex ) ) ).

% one_reorient
thf(fact_1196_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
      = ( X2 = one_one_real ) ) ).

% one_reorient
thf(fact_1197_one__reorient,axiom,
    ! [X2: rat] :
      ( ( one_one_rat = X2 )
      = ( X2 = one_one_rat ) ) ).

% one_reorient
thf(fact_1198_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_1199_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
      = ( X2 = one_one_int ) ) ).

% one_reorient
thf(fact_1200_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_1201_verit__la__generic,axiom,
    ! [A2: int,X2: int] :
      ( ( ord_less_eq_int @ A2 @ X2 )
      | ( A2 = X2 )
      | ( ord_less_eq_int @ X2 @ A2 ) ) ).

% verit_la_generic
thf(fact_1202_imp__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P4: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P4 ) ) ) ) ).

% imp_le_cong
thf(fact_1203_conj__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P4: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P4 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P4 ) ) ) ) ).

% conj_le_cong
thf(fact_1204_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_1205_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1206_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1207_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1208_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_1209_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_1210_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1211_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1212_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_1213_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1214_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1215_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1216_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1217_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B3: $o,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A2 )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B3 )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1218_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B3: $o,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A2 )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B3 )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1219_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_1220_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B3: $o] :
      ( ( A2
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A2
       => ( ( B3
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B3
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_1221_vebt__maxt_Osimps_I1_J,axiom,
    ! [B3: $o,A2: $o] :
      ( ( B3
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B3
       => ( ( A2
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_1222_vebt__succ_Osimps_I1_J,axiom,
    ! [B3: $o,Uu: $o] :
      ( ( B3
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B3
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_1223_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_1224_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_1225_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1226_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1227_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1228_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1229_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1230_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1240_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1241_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1242_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1243_reals__Archimedean2,axiom,
    ! [X2: rat] :
    ? [N3: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% reals_Archimedean2
thf(fact_1244_reals__Archimedean2,axiom,
    ! [X2: real] :
    ? [N3: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_1245_real__arch__simple,axiom,
    ! [X2: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_1246_real__arch__simple,axiom,
    ! [X2: rat] :
    ? [N3: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% real_arch_simple
thf(fact_1247_vebt__pred_Osimps_I3_J,axiom,
    ! [B3: $o,A2: $o,Va: nat] :
      ( ( B3
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B3
       => ( ( A2
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_1248_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1249_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1250_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1251_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1252_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1253_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_1254_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_1255_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_1256_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_1257_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_1258_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1259_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_1260_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_1261_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1262_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1263_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1264_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1265_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1266_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1267_Suc__inject,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y3 ) )
     => ( X2 = Y3 ) ) ).

% Suc_inject
thf(fact_1268_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1269_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1270_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1271_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1272_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1273_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1274_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1275_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X5: nat,Y4: nat] :
              ( ( P @ X5 @ Y4 )
             => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1276_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1277_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1278_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1279_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1280_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1281_vebt__buildup_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( X2
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va2: nat] :
              ( X2
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_1282_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1283_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1284_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1285_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1286_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1287_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1288_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1289_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ I4 ) ) ) ) ).

% All_less_Suc
thf(fact_1290_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1291_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1292_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1293_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1294_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1295_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1296_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_1297_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1298_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1299_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1300_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1301_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1302_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
     => ? [M4: nat] :
          ( M7
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1303_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1304_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1305_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1306_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1307_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1308_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z4: nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z4 )
               => ( R @ X5 @ Z4 ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1309_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1310_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1311_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1312_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1313_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1314_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1315_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1316_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1317_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1318_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1319_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1320_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1321_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1322_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1323_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1324_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1325_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1326_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1327_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1328_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1329_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_1330_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_1331_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_1332_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_1333_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1334_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M2: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1335_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1336_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1337_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1338_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1339_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1340_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1341_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1342_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1343_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1344_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1345_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1346_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1347_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1348_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B3: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A2 @ B3 ) ) ).

% vebt_delete.simps(3)
thf(fact_1349_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1350_linorder__neqE__linordered__idom,axiom,
    ! [X2: real,Y3: real] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_real @ X2 @ Y3 )
       => ( ord_less_real @ Y3 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1351_linorder__neqE__linordered__idom,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_rat @ X2 @ Y3 )
       => ( ord_less_rat @ Y3 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1352_linorder__neqE__linordered__idom,axiom,
    ! [X2: int,Y3: int] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_int @ X2 @ Y3 )
       => ( ord_less_int @ Y3 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1353_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1354_enumerate__step,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ S )
     => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ ( infini7641415182203889163d_enat @ S @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_1355_enumerate__step,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_1356_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_1357_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B3: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A2 @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_1358_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_1359_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_1360_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_1361_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( A2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_1362_option_Osize__gen_I1_J,axiom,
    ! [X2: nat > nat] :
      ( ( size_option_nat @ X2 @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1363_option_Osize__gen_I1_J,axiom,
    ! [X2: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X2 @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1364_option_Osize__gen_I1_J,axiom,
    ! [X2: num > nat] :
      ( ( size_option_num @ X2 @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1365_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1366_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1367_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1368_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1369_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1370_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1371_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X2
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_1372_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info @ ( suc @ ( suc @ N ) ) @ TreeList @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_1373_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% neg_int_cases
thf(fact_1374_frac__eq,axiom,
    ! [X2: real] :
      ( ( ( archim2898591450579166408c_real @ X2 )
        = X2 )
      = ( ( ord_less_eq_real @ zero_zero_real @ X2 )
        & ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% frac_eq
thf(fact_1375_frac__eq,axiom,
    ! [X2: rat] :
      ( ( ( archimedean_frac_rat @ X2 )
        = X2 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
        & ( ord_less_rat @ X2 @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_1376_finite__enum__subset,axiom,
    ! [X6: set_Extended_enat,Y7: set_Extended_enat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X6 ) )
         => ( ( infini7641415182203889163d_enat @ X6 @ I2 )
            = ( infini7641415182203889163d_enat @ Y7 @ I2 ) ) )
     => ( ( finite4001608067531595151d_enat @ X6 )
       => ( ( finite4001608067531595151d_enat @ Y7 )
         => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ X6 ) @ ( finite121521170596916366d_enat @ Y7 ) )
           => ( ord_le7203529160286727270d_enat @ X6 @ Y7 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_1377_finite__enum__subset,axiom,
    ! [X6: set_nat,Y7: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
         => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
            = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
     => ( ( finite_finite_nat @ X6 )
       => ( ( finite_finite_nat @ Y7 )
         => ( ( ord_less_eq_nat @ ( finite_card_nat @ X6 ) @ ( finite_card_nat @ Y7 ) )
           => ( ord_less_eq_set_nat @ X6 @ Y7 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_1378_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_1379_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_1380_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_1381_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_1382_add_Oinverse__inverse,axiom,
    ! [A2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1383_add_Oinverse__inverse,axiom,
    ! [A2: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1384_add_Oinverse__inverse,axiom,
    ! [A2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1385_add_Oinverse__inverse,axiom,
    ! [A2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1386_add_Oinverse__inverse,axiom,
    ! [A2: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1387_neg__equal__iff__equal,axiom,
    ! [A2: int,B3: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = ( uminus_uminus_int @ B3 ) )
      = ( A2 = B3 ) ) ).

% neg_equal_iff_equal
thf(fact_1388_neg__equal__iff__equal,axiom,
    ! [A2: real,B3: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = ( uminus_uminus_real @ B3 ) )
      = ( A2 = B3 ) ) ).

% neg_equal_iff_equal
thf(fact_1389_neg__equal__iff__equal,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = ( uminus_uminus_rat @ B3 ) )
      = ( A2 = B3 ) ) ).

% neg_equal_iff_equal
thf(fact_1390_neg__equal__iff__equal,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = ( uminus1351360451143612070nteger @ B3 ) )
      = ( A2 = B3 ) ) ).

% neg_equal_iff_equal
thf(fact_1391_neg__equal__iff__equal,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = ( uminus1482373934393186551omplex @ B3 ) )
      = ( A2 = B3 ) ) ).

% neg_equal_iff_equal
thf(fact_1392_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ A2 )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1393_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1394_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1395_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1396_diff__zero,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% diff_zero
thf(fact_1397_diff__zero,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_zero
thf(fact_1398_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_1399_diff__zero,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_zero
thf(fact_1400_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1401_diff__0__right,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% diff_0_right
thf(fact_1402_diff__0__right,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_0_right
thf(fact_1403_diff__0__right,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_0_right
thf(fact_1404_diff__self,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ A2 )
      = zero_zero_real ) ).

% diff_self
thf(fact_1405_diff__self,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1406_diff__self,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% diff_self
thf(fact_1407_div__by__0,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1408_div__by__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1409_div__by__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1410_div__by__0,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1411_div__by__0,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1412_div__0,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% div_0
thf(fact_1413_div__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% div_0
thf(fact_1414_div__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% div_0
thf(fact_1415_div__0,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ zero_zero_real @ A2 )
      = zero_zero_real ) ).

% div_0
thf(fact_1416_div__0,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A2 )
      = zero_zero_complex ) ).

% div_0
thf(fact_1417_neg__le__iff__le,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_eq_real @ A2 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_1418_neg__le__iff__le,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_1419_neg__le__iff__le,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_1420_neg__le__iff__le,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_1421_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = A2 )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_1422_neg__equal__zero,axiom,
    ! [A2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = A2 )
      = ( A2 = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_1423_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = A2 )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_1424_neg__equal__zero,axiom,
    ! [A2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = A2 )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_1425_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2
        = ( uminus_uminus_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_1426_equal__neg__zero,axiom,
    ! [A2: real] :
      ( ( A2
        = ( uminus_uminus_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_1427_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_1428_equal__neg__zero,axiom,
    ! [A2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_1429_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_1430_neg__equal__0__iff__equal,axiom,
    ! [A2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_1431_neg__equal__0__iff__equal,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_1432_neg__equal__0__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_1433_neg__equal__0__iff__equal,axiom,
    ! [A2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_1434_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A2 ) )
      = ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1435_neg__0__equal__iff__equal,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A2 ) )
      = ( zero_zero_real = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1436_neg__0__equal__iff__equal,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A2 ) )
      = ( zero_zero_rat = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1437_neg__0__equal__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A2 ) )
      = ( zero_z3403309356797280102nteger = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1438_neg__0__equal__iff__equal,axiom,
    ! [A2: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A2 ) )
      = ( zero_zero_complex = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1439_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_1440_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_1441_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_1442_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_1443_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_1444_neg__less__iff__less,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_1445_neg__less__iff__less,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_real @ A2 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_1446_neg__less__iff__less,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_1447_neg__less__iff__less,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_1448_minus__diff__eq,axiom,
    ! [A2: int,B3: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B3 ) )
      = ( minus_minus_int @ B3 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1449_minus__diff__eq,axiom,
    ! [A2: real,B3: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B3 ) )
      = ( minus_minus_real @ B3 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1450_minus__diff__eq,axiom,
    ! [A2: rat,B3: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B3 ) )
      = ( minus_minus_rat @ B3 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1451_minus__diff__eq,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B3 ) )
      = ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1452_minus__diff__eq,axiom,
    ! [A2: complex,B3: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B3 ) )
      = ( minus_minus_complex @ B3 @ A2 ) ) ).

% minus_diff_eq
thf(fact_1453_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1454_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1455_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1456_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1457_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1458_diff__ge__0__iff__ge,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B3 ) )
      = ( ord_less_eq_real @ B3 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1459_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B3 ) )
      = ( ord_less_eq_rat @ B3 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1460_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
      = ( ord_less_eq_int @ B3 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1461_diff__gt__0__iff__gt,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B3 ) )
      = ( ord_less_real @ B3 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1462_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B3 ) )
      = ( ord_less_rat @ B3 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1463_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
      = ( ord_less_int @ B3 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1464_neg__0__le__iff__le,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_1465_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_1466_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_1467_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_1468_neg__le__0__iff__le,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1469_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1470_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1471_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_1472_less__eq__neg__nonpos,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_1473_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_1474_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_1475_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_1476_neg__less__eq__nonneg,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1477_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1478_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1479_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_1480_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1481_neg__less__0__iff__less,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1482_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1483_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_1484_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_1485_neg__0__less__iff__less,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_1486_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_1487_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_1488_neg__less__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% neg_less_pos
thf(fact_1489_neg__less__pos,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% neg_less_pos
thf(fact_1490_neg__less__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_pos
thf(fact_1491_neg__less__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_pos
thf(fact_1492_less__neg__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_1493_less__neg__neg,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_1494_less__neg__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_1495_less__neg__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_1496_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1497_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1498_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1499_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1500_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% div_self
thf(fact_1501_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ A2 @ A2 )
        = one_one_int ) ) ).

% div_self
thf(fact_1502_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ A2 @ A2 )
        = one_one_nat ) ) ).

% div_self
thf(fact_1503_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ A2 )
        = one_one_real ) ) ).

% div_self
thf(fact_1504_div__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A2 @ A2 )
        = one_one_complex ) ) ).

% div_self
thf(fact_1505_verit__minus__simplify_I3_J,axiom,
    ! [B3: int] :
      ( ( minus_minus_int @ zero_zero_int @ B3 )
      = ( uminus_uminus_int @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_1506_verit__minus__simplify_I3_J,axiom,
    ! [B3: real] :
      ( ( minus_minus_real @ zero_zero_real @ B3 )
      = ( uminus_uminus_real @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_1507_verit__minus__simplify_I3_J,axiom,
    ! [B3: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B3 )
      = ( uminus_uminus_rat @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_1508_verit__minus__simplify_I3_J,axiom,
    ! [B3: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B3 )
      = ( uminus1351360451143612070nteger @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_1509_verit__minus__simplify_I3_J,axiom,
    ! [B3: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B3 )
      = ( uminus1482373934393186551omplex @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_1510_diff__0,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ zero_zero_int @ A2 )
      = ( uminus_uminus_int @ A2 ) ) ).

% diff_0
thf(fact_1511_diff__0,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ zero_zero_real @ A2 )
      = ( uminus_uminus_real @ A2 ) ) ).

% diff_0
thf(fact_1512_diff__0,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A2 )
      = ( uminus_uminus_rat @ A2 ) ) ).

% diff_0
thf(fact_1513_diff__0,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
      = ( uminus1351360451143612070nteger @ A2 ) ) ).

% diff_0
thf(fact_1514_diff__0,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A2 )
      = ( uminus1482373934393186551omplex @ A2 ) ) ).

% diff_0
thf(fact_1515_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1516_card_Oempty,axiom,
    ( ( finite_card_complex @ bot_bot_set_complex )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1517_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1518_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1519_card_Oempty,axiom,
    ( ( finite_card_real @ bot_bot_set_real )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1520_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1521_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1522_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1523_card_Oinfinite,axiom,
    ! [A3: set_list_nat] :
      ( ~ ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite_card_list_nat @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1524_card_Oinfinite,axiom,
    ! [A3: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite_card_set_nat @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1525_card_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_card_nat @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1526_card_Oinfinite,axiom,
    ! [A3: set_int] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( finite_card_int @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1527_card_Oinfinite,axiom,
    ! [A3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( finite_card_complex @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1528_card_Oinfinite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite711546835091564841at_nat @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1529_card_Oinfinite,axiom,
    ! [A3: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite121521170596916366d_enat @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1530_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1531_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1532_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1533_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_1534_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_1535_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_1536_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_1537_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_1538_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_1539_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_1540_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_1541_card__0__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( ( finite_card_list_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
thf(fact_1542_card__0__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( ( finite_card_set_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_1543_card__0__eq,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( finite_card_complex @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_complex ) ) ) ).

% card_0_eq
thf(fact_1544_card__0__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( finite711546835091564841at_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% card_0_eq
thf(fact_1545_card__0__eq,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( finite121521170596916366d_enat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bo7653980558646680370d_enat ) ) ) ).

% card_0_eq
thf(fact_1546_card__0__eq,axiom,
    ! [A3: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( finite_card_real @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_real ) ) ) ).

% card_0_eq
thf(fact_1547_card__0__eq,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( finite_card_o @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_1548_card__0__eq,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( finite_card_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_1549_card__0__eq,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( finite_card_int @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_1550_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_1551_finite__enumerate__mono__iff,axiom,
    ! [S: set_Extended_enat,M: nat,N: nat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ord_less_nat @ M @ ( finite121521170596916366d_enat @ S ) )
       => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
         => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) )
            = ( ord_less_nat @ M @ N ) ) ) ) ) ).

% finite_enumerate_mono_iff
thf(fact_1552_finite__enumerate__mono__iff,axiom,
    ! [S: set_nat,M: nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ M @ ( finite_card_nat @ S ) )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
         => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
            = ( ord_less_nat @ M @ N ) ) ) ) ) ).

% finite_enumerate_mono_iff
thf(fact_1553_diff__eq__diff__eq,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A2 @ B3 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A2 = B3 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1554_diff__eq__diff__eq,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A2 @ B3 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A2 = B3 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1555_diff__eq__diff__eq,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A2 @ B3 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A2 = B3 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1556_equation__minus__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( uminus_uminus_int @ B3 ) )
      = ( B3
        = ( uminus_uminus_int @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1557_equation__minus__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( A2
        = ( uminus_uminus_real @ B3 ) )
      = ( B3
        = ( uminus_uminus_real @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1558_equation__minus__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B3 ) )
      = ( B3
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1559_equation__minus__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ B3 ) )
      = ( B3
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1560_equation__minus__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ B3 ) )
      = ( B3
        = ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1561_minus__equation__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B3 )
      = ( ( uminus_uminus_int @ B3 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1562_minus__equation__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = B3 )
      = ( ( uminus_uminus_real @ B3 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1563_minus__equation__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B3 )
      = ( ( uminus_uminus_rat @ B3 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1564_minus__equation__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = B3 )
      = ( ( uminus1351360451143612070nteger @ B3 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1565_minus__equation__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = B3 )
      = ( ( uminus1482373934393186551omplex @ B3 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1566_minus__diff__commute,axiom,
    ! [B3: int,A2: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B3 ) @ A2 )
      = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).

% minus_diff_commute
thf(fact_1567_minus__diff__commute,axiom,
    ! [B3: real,A2: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B3 ) @ A2 )
      = ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ).

% minus_diff_commute
thf(fact_1568_minus__diff__commute,axiom,
    ! [B3: rat,A2: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B3 ) @ A2 )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ).

% minus_diff_commute
thf(fact_1569_minus__diff__commute,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ).

% minus_diff_commute
thf(fact_1570_minus__diff__commute,axiom,
    ! [B3: complex,A2: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A2 )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) ) ).

% minus_diff_commute
thf(fact_1571_diff__right__commute,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B3 )
      = ( minus_minus_real @ ( minus_minus_real @ A2 @ B3 ) @ C ) ) ).

% diff_right_commute
thf(fact_1572_diff__right__commute,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B3 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C ) ) ).

% diff_right_commute
thf(fact_1573_diff__right__commute,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B3 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C ) ) ).

% diff_right_commute
thf(fact_1574_diff__right__commute,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C ) ) ).

% diff_right_commute
thf(fact_1575_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1576_size__neq__size__imp__neq,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( ( size_size_VEBT_VEBT @ X2 )
       != ( size_size_VEBT_VEBT @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1577_size__neq__size__imp__neq,axiom,
    ! [X2: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X2 )
       != ( size_s6755466524823107622T_VEBT @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1578_size__neq__size__imp__neq,axiom,
    ! [X2: num,Y3: num] :
      ( ( ( size_size_num @ X2 )
       != ( size_size_num @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1579_size__neq__size__imp__neq,axiom,
    ! [X2: list_o,Y3: list_o] :
      ( ( ( size_size_list_o @ X2 )
       != ( size_size_list_o @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1580_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X2 )
       != ( size_size_list_nat @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1581_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1582_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1583_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1584_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_1585_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A2 @ B3 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A2 @ B3 )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1586_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A2 @ B3 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A2 @ B3 )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1587_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A2 @ B3 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A2 @ B3 )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1588_diff__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).

% diff_right_mono
thf(fact_1589_diff__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).

% diff_right_mono
thf(fact_1590_diff__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).

% diff_right_mono
thf(fact_1591_diff__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).

% diff_left_mono
thf(fact_1592_diff__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).

% diff_left_mono
thf(fact_1593_diff__left__mono,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).

% diff_left_mono
thf(fact_1594_diff__mono,axiom,
    ! [A2: real,B3: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% diff_mono
thf(fact_1595_diff__mono,axiom,
    ! [A2: rat,B3: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).

% diff_mono
thf(fact_1596_diff__mono,axiom,
    ! [A2: int,B3: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).

% diff_mono
thf(fact_1597_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z3: real] : Y6 = Z3 )
    = ( ^ [A4: real,B4: real] :
          ( ( minus_minus_real @ A4 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1598_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( minus_minus_rat @ A4 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1599_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
    = ( ^ [A4: int,B4: int] :
          ( ( minus_minus_int @ A4 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1600_diff__strict__mono,axiom,
    ! [A2: real,B3: real,D: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1601_diff__strict__mono,axiom,
    ! [A2: rat,B3: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1602_diff__strict__mono,axiom,
    ! [A2: int,B3: int,D: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1603_diff__eq__diff__less,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A2 @ B3 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A2 @ B3 )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1604_diff__eq__diff__less,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A2 @ B3 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A2 @ B3 )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1605_diff__eq__diff__less,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A2 @ B3 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A2 @ B3 )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1606_diff__strict__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).

% diff_strict_left_mono
thf(fact_1607_diff__strict__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).

% diff_strict_left_mono
thf(fact_1608_diff__strict__left__mono,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).

% diff_strict_left_mono
thf(fact_1609_diff__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1610_diff__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1611_diff__strict__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1612_le__imp__neg__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1613_le__imp__neg__le,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1614_le__imp__neg__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1615_le__imp__neg__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_1616_minus__le__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1617_minus__le__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1618_minus__le__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1619_minus__le__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_1620_le__minus__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B3 ) )
      = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1621_le__minus__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
      = ( ord_le3102999989581377725nteger @ B3 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1622_le__minus__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
      = ( ord_less_eq_rat @ B3 @ ( uminus_uminus_rat @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1623_le__minus__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B3 ) )
      = ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).

% le_minus_iff
thf(fact_1624_verit__negate__coefficient_I2_J,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_1625_verit__negate__coefficient_I2_J,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_1626_verit__negate__coefficient_I2_J,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_1627_verit__negate__coefficient_I2_J,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B3 )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_1628_less__minus__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B3 ) )
      = ( ord_less_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).

% less_minus_iff
thf(fact_1629_less__minus__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B3 ) )
      = ( ord_less_real @ B3 @ ( uminus_uminus_real @ A2 ) ) ) ).

% less_minus_iff
thf(fact_1630_less__minus__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
      = ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ A2 ) ) ) ).

% less_minus_iff
thf(fact_1631_less__minus__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
      = ( ord_le6747313008572928689nteger @ B3 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% less_minus_iff
thf(fact_1632_minus__less__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 )
      = ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_1633_minus__less__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B3 )
      = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_1634_minus__less__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_1635_minus__less__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_1636_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1637_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1638_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1639_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1640_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1641_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_1642_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
          = ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_1643_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1644_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1645_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1646_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1647_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1648_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_1649_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).

% vebt_member.simps(2)
thf(fact_1650_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_1651_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_1652_verit__less__mono__div__int2,axiom,
    ! [A3: int,B2: int,N: int] :
      ( ( ord_less_eq_int @ A3 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_1653_infinite__arbitrarily__large,axiom,
    ! [A3: set_list_nat,N: nat] :
      ( ~ ( finite8100373058378681591st_nat @ A3 )
     => ? [B8: set_list_nat] :
          ( ( finite8100373058378681591st_nat @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = N )
          & ( ord_le6045566169113846134st_nat @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1654_infinite__arbitrarily__large,axiom,
    ! [A3: set_set_nat,N: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A3 )
     => ? [B8: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = N )
          & ( ord_le6893508408891458716et_nat @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1655_infinite__arbitrarily__large,axiom,
    ! [A3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ? [B8: set_nat] :
          ( ( finite_finite_nat @ B8 )
          & ( ( finite_card_nat @ B8 )
            = N )
          & ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1656_infinite__arbitrarily__large,axiom,
    ! [A3: set_complex,N: nat] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ? [B8: set_complex] :
          ( ( finite3207457112153483333omplex @ B8 )
          & ( ( finite_card_complex @ B8 )
            = N )
          & ( ord_le211207098394363844omplex @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1657_infinite__arbitrarily__large,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( finite6177210948735845034at_nat @ B8 )
          & ( ( finite711546835091564841at_nat @ B8 )
            = N )
          & ( ord_le3146513528884898305at_nat @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1658_infinite__arbitrarily__large,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ? [B8: set_Extended_enat] :
          ( ( finite4001608067531595151d_enat @ B8 )
          & ( ( finite121521170596916366d_enat @ B8 )
            = N )
          & ( ord_le7203529160286727270d_enat @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1659_infinite__arbitrarily__large,axiom,
    ! [A3: set_int,N: nat] :
      ( ~ ( finite_finite_int @ A3 )
     => ? [B8: set_int] :
          ( ( finite_finite_int @ B8 )
          & ( ( finite_card_int @ B8 )
            = N )
          & ( ord_less_eq_set_int @ B8 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1660_card__subset__eq,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
       => ( ( ( finite_card_list_nat @ A3 )
            = ( finite_card_list_nat @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1661_card__subset__eq,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
       => ( ( ( finite_card_set_nat @ A3 )
            = ( finite_card_set_nat @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1662_card__subset__eq,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( ( ( finite_card_nat @ A3 )
            = ( finite_card_nat @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1663_card__subset__eq,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ( ( finite_card_complex @ A3 )
            = ( finite_card_complex @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1664_card__subset__eq,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
       => ( ( ( finite711546835091564841at_nat @ A3 )
            = ( finite711546835091564841at_nat @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1665_card__subset__eq,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ( ( finite121521170596916366d_enat @ A3 )
            = ( finite121521170596916366d_enat @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1666_card__subset__eq,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A3 @ B2 )
       => ( ( ( finite_card_int @ A3 )
            = ( finite_card_int @ B2 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_1667_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1668_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1669_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1670_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1671_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1672_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1673_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_real,R2: real > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A: real] :
            ( ( member_real @ A @ A3 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: real,A22: real,B: real] :
              ( ( member_real @ A1 @ A3 )
             => ( ( member_real @ A22 @ A3 )
               => ( ( member_real @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1674_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A3: set_real,R2: real > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A: real] :
            ( ( member_real @ A @ A3 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: real,A22: real,B: $o] :
              ( ( member_real @ A1 @ A3 )
             => ( ( member_real @ A22 @ A3 )
               => ( ( member_o @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1675_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_o,R2: $o > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A: $o] :
            ( ( member_o @ A @ A3 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: $o,A22: $o,B: real] :
              ( ( member_o @ A1 @ A3 )
             => ( ( member_o @ A22 @ A3 )
               => ( ( member_real @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1676_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A3: set_o,R2: $o > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A: $o] :
            ( ( member_o @ A @ A3 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: $o,A22: $o,B: $o] :
              ( ( member_o @ A1 @ A3 )
             => ( ( member_o @ A22 @ A3 )
               => ( ( member_o @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1677_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_complex,R2: complex > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A: complex] :
            ( ( member_complex @ A @ A3 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: complex,A22: complex,B: real] :
              ( ( member_complex @ A1 @ A3 )
             => ( ( member_complex @ A22 @ A3 )
               => ( ( member_real @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1678_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A3: set_complex,R2: complex > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A: complex] :
            ( ( member_complex @ A @ A3 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: complex,A22: complex,B: $o] :
              ( ( member_complex @ A1 @ A3 )
             => ( ( member_complex @ A22 @ A3 )
               => ( ( member_o @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1679_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_nat,R2: nat > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A: nat] :
            ( ( member_nat @ A @ A3 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: nat,A22: nat,B: real] :
              ( ( member_nat @ A1 @ A3 )
             => ( ( member_nat @ A22 @ A3 )
               => ( ( member_real @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1680_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A3: set_nat,R2: nat > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A: nat] :
            ( ( member_nat @ A @ A3 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: nat,A22: nat,B: $o] :
              ( ( member_nat @ A1 @ A3 )
             => ( ( member_nat @ A22 @ A3 )
               => ( ( member_o @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1681_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A3: set_int,R2: int > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A: int] :
            ( ( member_int @ A @ A3 )
           => ? [B9: real] :
                ( ( member_real @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: int,A22: int,B: real] :
              ( ( member_int @ A1 @ A3 )
             => ( ( member_int @ A22 @ A3 )
               => ( ( member_real @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1682_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A3: set_int,R2: int > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A: int] :
            ( ( member_int @ A @ A3 )
           => ? [B9: $o] :
                ( ( member_o @ B9 @ B2 )
                & ( R2 @ A @ B9 ) ) )
       => ( ! [A1: int,A22: int,B: $o] :
              ( ( member_int @ A1 @ A3 )
             => ( ( member_int @ A22 @ A3 )
               => ( ( member_o @ B @ B2 )
                 => ( ( R2 @ A1 @ B )
                   => ( ( R2 @ A22 @ B )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_1683_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_1684_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_1685_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_1686_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_1687_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_1688_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_1689_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_1690_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_1691_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_1692_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_1693_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_1694_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_1695_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_1696_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_1697_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_1698_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_1699_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_1700_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_1701_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_1702_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_1703_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_1704_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1705_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1706_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1707_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1708_diff__less__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1709_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1710_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1711_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_1712_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).

% vebt_member.simps(3)
thf(fact_1713_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_1714_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1715_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1716_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_1717_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( X2
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_1718_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y3: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y3 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y3 )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y3 )
         => ( ( ? [Uu2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y3 )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y3 )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y3 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_1719_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_1720_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_1721_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_1722_frac__ge__0,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) ) ).

% frac_ge_0
thf(fact_1723_frac__ge__0,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X2 ) ) ).

% frac_ge_0
thf(fact_1724_frac__lt__1,axiom,
    ! [X2: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X2 ) @ one_one_real ) ).

% frac_lt_1
thf(fact_1725_frac__lt__1,axiom,
    ! [X2: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X2 ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_1726_card__eq__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_list_nat )
        | ~ ( finite8100373058378681591st_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1727_card__eq__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_set_nat )
        | ~ ( finite1152437895449049373et_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1728_card__eq__0__iff,axiom,
    ! [A3: set_complex] :
      ( ( ( finite_card_complex @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_complex )
        | ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1729_card__eq__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1730_card__eq__0__iff,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ( finite121521170596916366d_enat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bo7653980558646680370d_enat )
        | ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1731_card__eq__0__iff,axiom,
    ! [A3: set_real] :
      ( ( ( finite_card_real @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_real )
        | ~ ( finite_finite_real @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1732_card__eq__0__iff,axiom,
    ! [A3: set_o] :
      ( ( ( finite_card_o @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_o )
        | ~ ( finite_finite_o @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1733_card__eq__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1734_card__eq__0__iff,axiom,
    ! [A3: set_int] :
      ( ( ( finite_card_int @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_int )
        | ~ ( finite_finite_int @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_1735_card__ge__0__finite,axiom,
    ! [A3: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
     => ( finite8100373058378681591st_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1736_card__ge__0__finite,axiom,
    ! [A3: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1737_card__ge__0__finite,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
     => ( finite_finite_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1738_card__ge__0__finite,axiom,
    ! [A3: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
     => ( finite_finite_int @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1739_card__ge__0__finite,axiom,
    ! [A3: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
     => ( finite3207457112153483333omplex @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1740_card__ge__0__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
     => ( finite6177210948735845034at_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1741_card__ge__0__finite,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A3 ) )
     => ( finite4001608067531595151d_enat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_1742_card__mono,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1743_card__mono,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1744_card__mono,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1745_card__mono,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ).

% card_mono
thf(fact_1746_card__mono,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1747_card__mono,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).

% card_mono
thf(fact_1748_card__mono,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A3 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ).

% card_mono
thf(fact_1749_card__seteq,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B2 ) @ ( finite_card_list_nat @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1750_card__seteq,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1751_card__seteq,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1752_card__seteq,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ B2 ) @ ( finite_card_complex @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1753_card__seteq,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B2 ) @ ( finite711546835091564841at_nat @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1754_card__seteq,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ B2 ) @ ( finite121521170596916366d_enat @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1755_card__seteq,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A3 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A3 ) )
         => ( A3 = B2 ) ) ) ) ).

% card_seteq
thf(fact_1756_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_list_nat,C2: nat] :
      ( ! [G: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G @ F2 )
         => ( ( finite8100373058378681591st_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C2 ) ) )
     => ( ( finite8100373058378681591st_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1757_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_set_nat,C2: nat] :
      ( ! [G: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G @ F2 )
         => ( ( finite1152437895449049373et_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C2 ) ) )
     => ( ( finite1152437895449049373et_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1758_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_nat,C2: nat] :
      ( ! [G: set_nat] :
          ( ( ord_less_eq_set_nat @ G @ F2 )
         => ( ( finite_finite_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
     => ( ( finite_finite_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1759_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_complex,C2: nat] :
      ( ! [G: set_complex] :
          ( ( ord_le211207098394363844omplex @ G @ F2 )
         => ( ( finite3207457112153483333omplex @ G )
           => ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C2 ) ) )
     => ( ( finite3207457112153483333omplex @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1760_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,C2: nat] :
      ( ! [G: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ G @ F2 )
         => ( ( finite6177210948735845034at_nat @ G )
           => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G ) @ C2 ) ) )
     => ( ( finite6177210948735845034at_nat @ F2 )
        & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1761_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_Extended_enat,C2: nat] :
      ( ! [G: set_Extended_enat] :
          ( ( ord_le7203529160286727270d_enat @ G @ F2 )
         => ( ( finite4001608067531595151d_enat @ G )
           => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ G ) @ C2 ) ) )
     => ( ( finite4001608067531595151d_enat @ F2 )
        & ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1762_finite__if__finite__subsets__card__bdd,axiom,
    ! [F2: set_int,C2: nat] :
      ( ! [G: set_int] :
          ( ( ord_less_eq_set_int @ G @ F2 )
         => ( ( finite_finite_int @ G )
           => ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C2 ) ) )
     => ( ( finite_finite_int @ F2 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C2 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_1763_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_list_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S ) )
     => ~ ! [T4: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T4 @ S )
           => ( ( ( finite_card_list_nat @ T4 )
                = N )
             => ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1764_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S ) )
     => ~ ! [T4: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T4 @ S )
           => ( ( ( finite_card_set_nat @ T4 )
                = N )
             => ~ ( finite1152437895449049373et_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1765_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
     => ~ ! [T4: set_nat] :
            ( ( ord_less_eq_set_nat @ T4 @ S )
           => ( ( ( finite_card_nat @ T4 )
                = N )
             => ~ ( finite_finite_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1766_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_complex] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S ) )
     => ~ ! [T4: set_complex] :
            ( ( ord_le211207098394363844omplex @ T4 @ S )
           => ( ( ( finite_card_complex @ T4 )
                = N )
             => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1767_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ S ) )
     => ~ ! [T4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ T4 @ S )
           => ( ( ( finite711546835091564841at_nat @ T4 )
                = N )
             => ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1768_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_Extended_enat] :
      ( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ S ) )
     => ~ ! [T4: set_Extended_enat] :
            ( ( ord_le7203529160286727270d_enat @ T4 @ S )
           => ( ( ( finite121521170596916366d_enat @ T4 )
                = N )
             => ~ ( finite4001608067531595151d_enat @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1769_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_int] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
     => ~ ! [T4: set_int] :
            ( ( ord_less_eq_set_int @ T4 @ S )
           => ( ( ( finite_card_int @ T4 )
                = N )
             => ~ ( finite_finite_int @ T4 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_1770_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_1771_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_1772_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_1773_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_1774_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_1775_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_1776_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_1777_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_1778_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_1779_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_1780_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_1781_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_1782_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_1783_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_1784_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_1785_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_1786_psubset__card__mono,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
       => ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1787_psubset__card__mono,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_less_set_set_nat @ A3 @ B2 )
       => ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1788_psubset__card__mono,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_set_nat @ A3 @ B2 )
       => ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1789_psubset__card__mono,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_set_int @ A3 @ B2 )
       => ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1790_psubset__card__mono,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_less_set_complex @ A3 @ B2 )
       => ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1791_psubset__card__mono,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le7866589430770878221at_nat @ A3 @ B2 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1792_psubset__card__mono,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le2529575680413868914d_enat @ A3 @ B2 )
       => ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_1793_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1794_finite__enumerate__in__set,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
       => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ S ) ) ) ).

% finite_enumerate_in_set
thf(fact_1795_finite__enumerate__in__set,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
       => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).

% finite_enumerate_in_set
thf(fact_1796_finite__enumerate__Ex,axiom,
    ! [S: set_Extended_enat,S2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( member_Extended_enat @ S2 @ S )
       => ? [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( finite121521170596916366d_enat @ S ) )
            & ( ( infini7641415182203889163d_enat @ S @ N3 )
              = S2 ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_1797_finite__enumerate__Ex,axiom,
    ! [S: set_nat,S2: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( member_nat @ S2 @ S )
       => ? [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
            & ( ( infini8530281810654367211te_nat @ S @ N3 )
              = S2 ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_1798_finite__enum__ext,axiom,
    ! [X6: set_Extended_enat,Y7: set_Extended_enat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X6 ) )
         => ( ( infini7641415182203889163d_enat @ X6 @ I2 )
            = ( infini7641415182203889163d_enat @ Y7 @ I2 ) ) )
     => ( ( finite4001608067531595151d_enat @ X6 )
       => ( ( finite4001608067531595151d_enat @ Y7 )
         => ( ( ( finite121521170596916366d_enat @ X6 )
              = ( finite121521170596916366d_enat @ Y7 ) )
           => ( X6 = Y7 ) ) ) ) ) ).

% finite_enum_ext
thf(fact_1799_finite__enum__ext,axiom,
    ! [X6: set_nat,Y7: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
         => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
            = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
     => ( ( finite_finite_nat @ X6 )
       => ( ( finite_finite_nat @ Y7 )
         => ( ( ( finite_card_nat @ X6 )
              = ( finite_card_nat @ Y7 ) )
           => ( X6 = Y7 ) ) ) ) ) ).

% finite_enum_ext
thf(fact_1800_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_1801_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).

% vebt_member.simps(4)
thf(fact_1802_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_1803_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_1804_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% nonpos_int_cases
thf(fact_1805_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_1806_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_1807_card__gt__0__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_list_nat )
        & ( finite8100373058378681591st_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1808_card__gt__0__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_set_nat )
        & ( finite1152437895449049373et_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1809_card__gt__0__iff,axiom,
    ! [A3: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
      = ( ( A3 != bot_bot_set_complex )
        & ( finite3207457112153483333omplex @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1810_card__gt__0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
      = ( ( A3 != bot_bo2099793752762293965at_nat )
        & ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1811_card__gt__0__iff,axiom,
    ! [A3: set_Extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A3 ) )
      = ( ( A3 != bot_bo7653980558646680370d_enat )
        & ( finite4001608067531595151d_enat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1812_card__gt__0__iff,axiom,
    ! [A3: set_real] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
      = ( ( A3 != bot_bot_set_real )
        & ( finite_finite_real @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1813_card__gt__0__iff,axiom,
    ! [A3: set_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
      = ( ( A3 != bot_bot_set_o )
        & ( finite_finite_o @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1814_card__gt__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_nat )
        & ( finite_finite_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1815_card__gt__0__iff,axiom,
    ! [A3: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
      = ( ( A3 != bot_bot_set_int )
        & ( finite_finite_int @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_1816_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: list_nat] :
              ( ( member_list_nat @ X @ A3 )
             => ! [Y: list_nat] :
                  ( ( member_list_nat @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1817_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: set_nat] :
              ( ( member_set_nat @ X @ A3 )
             => ! [Y: set_nat] :
                  ( ( member_set_nat @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1818_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A3 )
             => ! [Y: nat] :
                  ( ( member_nat @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1819_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ A3 )
             => ! [Y: int] :
                  ( ( member_int @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1820_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ A3 )
             => ! [Y: complex] :
                  ( ( member_complex @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1821_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
             => ! [Y: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1822_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: extended_enat] :
              ( ( member_Extended_enat @ X @ A3 )
             => ! [Y: extended_enat] :
                  ( ( member_Extended_enat @ Y @ A3 )
                 => ( X = Y ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_1823_card__psubset,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_le1190675801316882794st_nat @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1824_card__psubset,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_set_set_nat @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1825_card__psubset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_set_nat @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1826_card__psubset,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_set_complex @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1827_card__psubset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
         => ( ord_le7866589430770878221at_nat @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1828_card__psubset,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_le2529575680413868914d_enat @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1829_card__psubset,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A3 @ B2 )
       => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_set_int @ A3 @ B2 ) ) ) ) ).

% card_psubset
thf(fact_1830_finite__enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_Extended_enat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( finite4001608067531595151d_enat @ S )
       => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
         => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ) ).

% finite_enumerate_mono
thf(fact_1831_finite__enumerate__mono,axiom,
    ! [M: nat,N: nat,S: set_nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( finite_finite_nat @ S )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
         => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).

% finite_enumerate_mono
thf(fact_1832_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1833_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_1834_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
       => ~ ! [N3: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% int_cases3
thf(fact_1835_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_1836_negD,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ zero_zero_int )
     => ? [N3: nat] :
          ( X2
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_1837_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_1838_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_1839_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_1840_finite__le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_1841_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E2 )
     => ~ ! [N3: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_1842_nat__approx__posE,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ~ ! [N3: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_1843_finite__enumerate__step,axiom,
    ! [S: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ S ) )
       => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ ( infini7641415182203889163d_enat @ S @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_1844_finite__enumerate__step,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_1845_le__divide__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
        = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1846_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
        = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1847_le__divide__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
        = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1848_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
        = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1849_divide__le__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
        = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1850_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
        = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1851_divide__le__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
        = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1852_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
        = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1853_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_divide_1_iff
thf(fact_1854_zero__less__divide__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% zero_less_divide_1_iff
thf(fact_1855_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
        = ( ord_less_rat @ A2 @ B3 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1856_less__divide__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
        = ( ord_less_real @ A2 @ B3 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1857_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
        = ( ord_less_rat @ B3 @ A2 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1858_less__divide__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
        = ( ord_less_real @ B3 @ A2 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1859_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
        = ( ord_less_rat @ B3 @ A2 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1860_divide__less__eq__1__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
        = ( ord_less_real @ B3 @ A2 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1861_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
        = ( ord_less_rat @ A2 @ B3 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1862_divide__less__eq__1__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
        = ( ord_less_real @ A2 @ B3 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1863_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_1864_divide__less__0__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_1865_Diff__empty,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
      = A3 ) ).

% Diff_empty
thf(fact_1866_Diff__empty,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Diff_empty
thf(fact_1867_Diff__empty,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Diff_empty
thf(fact_1868_Diff__empty,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Diff_empty
thf(fact_1869_empty__Diff,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_1870_empty__Diff,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1871_empty__Diff,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1872_empty__Diff,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1873_Diff__cancel,axiom,
    ! [A3: set_real] :
      ( ( minus_minus_set_real @ A3 @ A3 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_1874_Diff__cancel,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ A3 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1875_Diff__cancel,axiom,
    ! [A3: set_int] :
      ( ( minus_minus_set_int @ A3 @ A3 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1876_Diff__cancel,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ A3 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1877_finite__Diff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1878_finite__Diff,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1879_finite__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1880_finite__Diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1881_finite__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_1882_finite__Diff2,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) )
        = ( finite_finite_int @ A3 ) ) ) ).

% finite_Diff2
thf(fact_1883_finite__Diff2,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
        = ( finite3207457112153483333omplex @ A3 ) ) ) ).

% finite_Diff2
thf(fact_1884_finite__Diff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
        = ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% finite_Diff2
thf(fact_1885_finite__Diff2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
        = ( finite4001608067531595151d_enat @ A3 ) ) ) ).

% finite_Diff2
thf(fact_1886_finite__Diff2,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
        = ( finite_finite_nat @ A3 ) ) ) ).

% finite_Diff2
thf(fact_1887_Compl__anti__mono,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).

% Compl_anti_mono
thf(fact_1888_Compl__subset__Compl__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( uminus1532241313380277803et_int @ B2 ) )
      = ( ord_less_eq_set_int @ B2 @ A3 ) ) ).

% Compl_subset_Compl_iff
thf(fact_1889_divide__eq__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A2 @ B3 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1890_divide__eq__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( divide_divide_real @ A2 @ B3 )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1891_divide__eq__0__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        | ( B3 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1892_divide__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ( divide_divide_rat @ C @ A2 )
        = ( divide_divide_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_left
thf(fact_1893_divide__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ( divide_divide_real @ C @ A2 )
        = ( divide_divide_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_left
thf(fact_1894_divide__cancel__left,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A2 )
        = ( divide1717551699836669952omplex @ C @ B3 ) )
      = ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_left
thf(fact_1895_divide__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A2 @ C )
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_right
thf(fact_1896_divide__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ( divide_divide_real @ A2 @ C )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_right
thf(fact_1897_divide__cancel__right,axiom,
    ! [A2: complex,C: complex,B3: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ C )
        = ( divide1717551699836669952omplex @ B3 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% divide_cancel_right
thf(fact_1898_division__ring__divide__zero,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1899_division__ring__divide__zero,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1900_division__ring__divide__zero,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1901_Diff__eq__empty__iff,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ( minus_minus_set_real @ A3 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A3 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1902_Diff__eq__empty__iff,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A3 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1903_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1904_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_1905_divide__eq__1__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A2 @ B3 )
        = one_one_rat )
      = ( ( B3 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% divide_eq_1_iff
thf(fact_1906_divide__eq__1__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( divide_divide_real @ A2 @ B3 )
        = one_one_real )
      = ( ( B3 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% divide_eq_1_iff
thf(fact_1907_divide__eq__1__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
        = one_one_complex )
      = ( ( B3 != zero_zero_complex )
        & ( A2 = B3 ) ) ) ).

% divide_eq_1_iff
thf(fact_1908_one__eq__divide__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A2 @ B3 ) )
      = ( ( B3 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% one_eq_divide_iff
thf(fact_1909_one__eq__divide__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A2 @ B3 ) )
      = ( ( B3 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% one_eq_divide_iff
thf(fact_1910_one__eq__divide__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A2 @ B3 ) )
      = ( ( B3 != zero_zero_complex )
        & ( A2 = B3 ) ) ) ).

% one_eq_divide_iff
thf(fact_1911_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1912_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ A2 )
        = one_one_real ) ) ).

% divide_self
thf(fact_1913_divide__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A2 @ A2 )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1914_divide__self__if,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( ( divide_divide_rat @ A2 @ A2 )
          = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( divide_divide_rat @ A2 @ A2 )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_1915_divide__self__if,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( ( divide_divide_real @ A2 @ A2 )
          = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( ( divide_divide_real @ A2 @ A2 )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_1916_divide__self__if,axiom,
    ! [A2: complex] :
      ( ( ( A2 = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A2 @ A2 )
          = zero_zero_complex ) )
      & ( ( A2 != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A2 @ A2 )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_1917_divide__eq__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B3 @ A2 )
        = one_one_rat )
      = ( ( A2 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% divide_eq_eq_1
thf(fact_1918_divide__eq__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( ( divide_divide_real @ B3 @ A2 )
        = one_one_real )
      = ( ( A2 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% divide_eq_eq_1
thf(fact_1919_eq__divide__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B3 @ A2 ) )
      = ( ( A2 != zero_zero_rat )
        & ( A2 = B3 ) ) ) ).

% eq_divide_eq_1
thf(fact_1920_eq__divide__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B3 @ A2 ) )
      = ( ( A2 != zero_zero_real )
        & ( A2 = B3 ) ) ) ).

% eq_divide_eq_1
thf(fact_1921_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_1922_one__divide__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( divide_divide_real @ one_one_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_1923_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_1924_zero__eq__1__divide__iff,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_1925_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% zle_diff1_eq
thf(fact_1926_zero__le__divide__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% zero_le_divide_1_iff
thf(fact_1927_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_divide_1_iff
thf(fact_1928_divide__le__0__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_1929_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_1930_Diff__infinite__finite,axiom,
    ! [T2: set_int,S: set_int] :
      ( ( finite_finite_int @ T2 )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1931_Diff__infinite__finite,axiom,
    ! [T2: set_complex,S: set_complex] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ~ ( finite3207457112153483333omplex @ S )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1932_Diff__infinite__finite,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ T2 )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1933_Diff__infinite__finite,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ~ ( finite4001608067531595151d_enat @ S )
       => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1934_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1935_Diff__mono,axiom,
    ! [A3: set_nat,C2: set_nat,D4: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ D4 @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1936_Diff__mono,axiom,
    ! [A3: set_int,C2: set_int,D4: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ D4 @ B2 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ ( minus_minus_set_int @ C2 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1937_Diff__subset,axiom,
    ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ A3 ) ).

% Diff_subset
thf(fact_1938_Diff__subset,axiom,
    ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ A3 ) ).

% Diff_subset
thf(fact_1939_double__diff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_1940_double__diff,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C2 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_1941_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ord_less_set_real @ A3 @ B2 )
     => ? [B: real] : ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1942_psubset__imp__ex__mem,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ord_less_set_o @ A3 @ B2 )
     => ? [B: $o] : ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1943_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ B2 )
     => ? [B: set_nat] : ( member_set_nat @ B @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1944_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A3 @ B2 )
     => ? [B: int] : ( member_int @ B @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1945_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B2 )
     => ? [B: nat] : ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1946_subset__Compl__self__eq,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
      = ( A3 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_1947_subset__Compl__self__eq,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_1948_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_1949_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_1950_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_1951_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_1952_card__less__sym__Diff,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1953_card__less__sym__Diff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1954_card__less__sym__Diff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1955_card__less__sym__Diff,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1956_card__less__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1957_card__less__sym__Diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1958_card__less__sym__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_1959_card__le__sym__Diff,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1960_card__le__sym__Diff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1961_card__le__sym__Diff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1962_card__le__sym__Diff,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1963_card__le__sym__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1964_card__le__sym__Diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1965_card__le__sym__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_1966_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1967_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1968_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).

% linordered_field_no_lb
thf(fact_1969_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).

% linordered_field_no_lb
thf(fact_1970_card__Diff__subset,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1971_card__Diff__subset,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ A3 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1972_card__Diff__subset,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1973_card__Diff__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1974_card__Diff__subset,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
       => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1975_card__Diff__subset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1976_card__Diff__subset,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A3 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).

% card_Diff_subset
thf(fact_1977_diff__card__le__card__Diff,axiom,
    ! [B2: set_list_nat,A3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1978_diff__card__le__card__Diff,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1979_diff__card__le__card__Diff,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1980_diff__card__le__card__Diff,axiom,
    ! [B2: set_complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1981_diff__card__le__card__Diff,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1982_diff__card__le__card__Diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1983_diff__card__le__card__Diff,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_1984_int__ops_I6_J,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% int_ops(6)
thf(fact_1985_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_1986_zdiff__int__split,axiom,
    ! [P: int > $o,X2: nat,Y3: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y3 ) ) )
      = ( ( ( ord_less_eq_nat @ Y3 @ X2 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
        & ( ( ord_less_nat @ X2 @ Y3 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_1987_divide__right__mono__neg,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1988_divide__right__mono__neg,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( divide_divide_rat @ A2 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1989_divide__nonpos__nonpos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1990_divide__nonpos__nonpos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1991_divide__nonpos__nonneg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1992_divide__nonpos__nonneg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1993_divide__nonneg__nonpos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1994_divide__nonneg__nonpos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1995_divide__nonneg__nonneg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1996_divide__nonneg__nonneg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1997_zero__le__divide__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1998_zero__le__divide__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1999_divide__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2000_divide__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2001_divide__le__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).

% divide_le_0_iff
thf(fact_2002_divide__le__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% divide_le_0_iff
thf(fact_2003_divide__neg__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2004_divide__neg__neg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2005_divide__neg__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2006_divide__neg__pos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2007_divide__pos__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2008_divide__pos__neg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2009_divide__pos__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2010_divide__pos__pos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2011_divide__less__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% divide_less_0_iff
thf(fact_2012_divide__less__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A2 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).

% divide_less_0_iff
thf(fact_2013_divide__less__cancel,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ A2 ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2014_divide__less__cancel,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ A2 ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2015_zero__less__divide__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2016_zero__less__divide__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2017_divide__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2018_divide__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2019_divide__strict__right__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2020_divide__strict__right__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2021_right__inverse__eq,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A2 @ B3 )
          = one_one_rat )
        = ( A2 = B3 ) ) ) ).

% right_inverse_eq
thf(fact_2022_right__inverse__eq,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( ( divide_divide_real @ A2 @ B3 )
          = one_one_real )
        = ( A2 = B3 ) ) ) ).

% right_inverse_eq
thf(fact_2023_right__inverse__eq,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
          = one_one_complex )
        = ( A2 = B3 ) ) ) ).

% right_inverse_eq
thf(fact_2024_nonzero__minus__divide__divide,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B3 ) )
        = ( divide_divide_real @ A2 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_2025_nonzero__minus__divide__divide,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B3 ) )
        = ( divide_divide_rat @ A2 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_2026_nonzero__minus__divide__divide,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) )
        = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_2027_nonzero__minus__divide__right,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) )
        = ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_2028_nonzero__minus__divide__right,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) )
        = ( divide_divide_rat @ A2 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_2029_nonzero__minus__divide__right,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
        = ( divide1717551699836669952omplex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_2030_frac__le,axiom,
    ! [Y3: real,X2: real,W2: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_2031_frac__le,axiom,
    ! [Y3: rat,X2: rat,W2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X2 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_2032_frac__less,axiom,
    ! [X2: real,Y3: real,W2: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_2033_frac__less,axiom,
    ! [X2: rat,Y3: rat,W2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ X2 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_2034_frac__less2,axiom,
    ! [X2: real,Y3: real,W2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_real @ W2 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_2035_frac__less2,axiom,
    ! [X2: rat,Y3: rat,W2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_rat @ W2 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_2036_divide__le__cancel,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).

% divide_le_cancel
thf(fact_2037_divide__le__cancel,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).

% divide_le_cancel
thf(fact_2038_divide__nonneg__neg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2039_divide__nonneg__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2040_divide__nonneg__pos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2041_divide__nonneg__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2042_divide__nonpos__neg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2043_divide__nonpos__neg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2044_divide__nonpos__pos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2045_divide__nonpos__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2046_divide__less__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B3 @ A2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ A2 @ B3 ) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2047_divide__less__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ B3 @ A2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ A2 @ B3 ) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2048_less__divide__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ A2 @ B3 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).

% less_divide_eq_1
thf(fact_2049_less__divide__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ A2 @ B3 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ B3 @ A2 ) ) ) ) ).

% less_divide_eq_1
thf(fact_2050_divide__eq__minus__1__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( divide_divide_real @ A2 @ B3 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B3 != zero_zero_real )
        & ( A2
          = ( uminus_uminus_real @ B3 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_2051_divide__eq__minus__1__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A2 @ B3 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B3 != zero_zero_rat )
        & ( A2
          = ( uminus_uminus_rat @ B3 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_2052_divide__eq__minus__1__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B3 != zero_zero_complex )
        & ( A2
          = ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_2053_divide__le__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B3 @ A2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ A2 @ B3 ) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2054_divide__le__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B3 @ A2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ A2 @ B3 ) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2055_le__divide__eq__1,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ A2 @ B3 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).

% le_divide_eq_1
thf(fact_2056_le__divide__eq__1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ A2 @ B3 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).

% le_divide_eq_1
thf(fact_2057_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_2058_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_2059_div__eq__minus1,axiom,
    ! [B3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_2060_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_2061_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_2062_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_2063_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_2064_compl__le__compl__iff,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
      = ( ord_less_eq_set_int @ Y3 @ X2 ) ) ).

% compl_le_compl_iff
thf(fact_2065_bits__div__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2066_bits__div__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2067_DiffI,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ A3 )
     => ( ~ ( member_real @ C @ B2 )
       => ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ) ).

% DiffI
thf(fact_2068_DiffI,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ A3 )
     => ( ~ ( member_o @ C @ B2 )
       => ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ) ).

% DiffI
thf(fact_2069_DiffI,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A3 )
     => ( ~ ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ).

% DiffI
thf(fact_2070_DiffI,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ A3 )
     => ( ~ ( member_int @ C @ B2 )
       => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).

% DiffI
thf(fact_2071_DiffI,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( ~ ( member_nat @ C @ B2 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).

% DiffI
thf(fact_2072_Diff__iff,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
      = ( ( member_real @ C @ A3 )
        & ~ ( member_real @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_2073_Diff__iff,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
      = ( ( member_o @ C @ A3 )
        & ~ ( member_o @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_2074_Diff__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
      = ( ( member_set_nat @ C @ A3 )
        & ~ ( member_set_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_2075_Diff__iff,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
      = ( ( member_int @ C @ A3 )
        & ~ ( member_int @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_2076_Diff__iff,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
      = ( ( member_nat @ C @ A3 )
        & ~ ( member_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_2077_Diff__idemp,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ B2 )
      = ( minus_minus_set_nat @ A3 @ B2 ) ) ).

% Diff_idemp
thf(fact_2078_ComplI,axiom,
    ! [C: real,A3: set_real] :
      ( ~ ( member_real @ C @ A3 )
     => ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) ) ) ).

% ComplI
thf(fact_2079_ComplI,axiom,
    ! [C: $o,A3: set_o] :
      ( ~ ( member_o @ C @ A3 )
     => ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) ) ) ).

% ComplI
thf(fact_2080_ComplI,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A3 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) ) ) ).

% ComplI
thf(fact_2081_ComplI,axiom,
    ! [C: nat,A3: set_nat] :
      ( ~ ( member_nat @ C @ A3 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).

% ComplI
thf(fact_2082_ComplI,axiom,
    ! [C: int,A3: set_int] :
      ( ~ ( member_int @ C @ A3 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).

% ComplI
thf(fact_2083_Compl__iff,axiom,
    ! [C: real,A3: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
      = ( ~ ( member_real @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_2084_Compl__iff,axiom,
    ! [C: $o,A3: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
      = ( ~ ( member_o @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_2085_Compl__iff,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
      = ( ~ ( member_set_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_2086_Compl__iff,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
      = ( ~ ( member_nat @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_2087_Compl__iff,axiom,
    ! [C: int,A3: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
      = ( ~ ( member_int @ C @ A3 ) ) ) ).

% Compl_iff
thf(fact_2088_bits__div__by__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2089_bits__div__by__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2090_real__of__nat__div3,axiom,
    ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_2091_DiffE,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
     => ~ ( ( member_real @ C @ A3 )
         => ( member_real @ C @ B2 ) ) ) ).

% DiffE
thf(fact_2092_DiffE,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
     => ~ ( ( member_o @ C @ A3 )
         => ( member_o @ C @ B2 ) ) ) ).

% DiffE
thf(fact_2093_DiffE,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
     => ~ ( ( member_set_nat @ C @ A3 )
         => ( member_set_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_2094_DiffE,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
     => ~ ( ( member_int @ C @ A3 )
         => ( member_int @ C @ B2 ) ) ) ).

% DiffE
thf(fact_2095_DiffE,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
     => ~ ( ( member_nat @ C @ A3 )
         => ( member_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_2096_ComplD,axiom,
    ! [C: real,A3: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
     => ~ ( member_real @ C @ A3 ) ) ).

% ComplD
thf(fact_2097_ComplD,axiom,
    ! [C: $o,A3: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
     => ~ ( member_o @ C @ A3 ) ) ).

% ComplD
thf(fact_2098_ComplD,axiom,
    ! [C: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
     => ~ ( member_set_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_2099_ComplD,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
     => ~ ( member_nat @ C @ A3 ) ) ).

% ComplD
thf(fact_2100_ComplD,axiom,
    ! [C: int,A3: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
     => ~ ( member_int @ C @ A3 ) ) ).

% ComplD
thf(fact_2101_DiffD1,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
     => ( member_real @ C @ A3 ) ) ).

% DiffD1
thf(fact_2102_DiffD1,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
     => ( member_o @ C @ A3 ) ) ).

% DiffD1
thf(fact_2103_DiffD1,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
     => ( member_set_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_2104_DiffD1,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
     => ( member_int @ C @ A3 ) ) ).

% DiffD1
thf(fact_2105_DiffD1,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
     => ( member_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_2106_DiffD2,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
     => ~ ( member_real @ C @ B2 ) ) ).

% DiffD2
thf(fact_2107_DiffD2,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
     => ~ ( member_o @ C @ B2 ) ) ).

% DiffD2
thf(fact_2108_DiffD2,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
     => ~ ( member_set_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_2109_DiffD2,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
     => ~ ( member_int @ C @ B2 ) ) ).

% DiffD2
thf(fact_2110_DiffD2,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
     => ~ ( member_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_2111_real__of__nat__div4,axiom,
    ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% real_of_nat_div4
thf(fact_2112_real__of__nat__div2,axiom,
    ! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).

% real_of_nat_div2
thf(fact_2113_subrelI,axiom,
    ! [R2: set_Pr6588086440996610945on_nat,S2: set_Pr6588086440996610945on_nat] :
      ( ! [X5: option_nat,Y4: option_nat] :
          ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X5 @ Y4 ) @ R2 )
         => ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le6406482658798684961on_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_2114_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_2115_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_2116_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_2117_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y4: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).

% subrelI
thf(fact_2118_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_2119_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_2120_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_2121_vebt__mint_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A: $o,B: $o] :
          ( X2
         != ( vEBT_Leaf @ A @ B ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_2122_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_2123_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_2124_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_2125_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_2126_compl__le__swap2,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X2 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) ) ).

% compl_le_swap2
thf(fact_2127_compl__le__swap1,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) )
     => ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).

% compl_le_swap1
thf(fact_2128_compl__mono,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X2 ) ) ) ).

% compl_mono
thf(fact_2129_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2130_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_2131_vebt__mint_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( A
                 => ( Y3
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A
                 => ( ( B
                     => ( Y3
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y3
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2132_vebt__maxt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( B
                 => ( Y3
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B
                 => ( ( A
                     => ( Y3
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y3
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2133_diff__shunt__var,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( ( minus_minus_set_real @ X2 @ Y3 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_2134_diff__shunt__var,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( ( minus_minus_set_o @ X2 @ Y3 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_2135_diff__shunt__var,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( minus_minus_set_nat @ X2 @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_2136_diff__shunt__var,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ( minus_minus_set_int @ X2 @ Y3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_2137_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_2138_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_2139_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2140_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_2141_pos__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int )
        = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_2142_neg__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_2143_div__neg__pos__less0,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_2144_div__positive,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% div_positive
thf(fact_2145_div__positive,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_int @ B3 @ A2 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% div_positive
thf(fact_2146_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ B3 )
       => ( ( divide_divide_nat @ A2 @ B3 )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2147_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ B3 )
       => ( ( divide_divide_int @ A2 @ B3 )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2148_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_2149_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M2 @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_2150_zdiv__mono1,axiom,
    ! [A2: int,A7: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ A7 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ ( divide_divide_int @ A7 @ B3 ) ) ) ) ).

% zdiv_mono1
thf(fact_2151_zdiv__mono2,axiom,
    ! [A2: int,B7: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B7 )
       => ( ( ord_less_eq_int @ B7 @ B3 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ ( divide_divide_int @ A2 @ B7 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_2152_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_2153_zdiv__mono1__neg,axiom,
    ! [A2: int,A7: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ A7 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B3 ) @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_2154_zdiv__mono2__neg,axiom,
    ! [A2: int,B7: int,B3: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B7 )
       => ( ( ord_less_eq_int @ B7 @ B3 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B7 ) @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_2155_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_2156_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_2157_div__nonneg__neg__le0,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_2158_div__nonpos__pos__le0,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_2159_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_2160_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
        = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_2161_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_2162_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
        = ( ( ord_less_eq_int @ B3 @ A2 )
          & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_2163_int__div__less__self,axiom,
    ! [X2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).

% int_div_less_self
thf(fact_2164_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_2165_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 )
       => ( ! [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_2166_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_2167_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_2168_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_2169_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_2170_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_2171_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_2172_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_2173_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_2174_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_2175_prod__decode__aux_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_2176_insert__absorb2,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
      = ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ).

% insert_absorb2
thf(fact_2177_insert__absorb2,axiom,
    ! [X2: real,A3: set_real] :
      ( ( insert_real @ X2 @ ( insert_real @ X2 @ A3 ) )
      = ( insert_real @ X2 @ A3 ) ) ).

% insert_absorb2
thf(fact_2178_insert__absorb2,axiom,
    ! [X2: $o,A3: set_o] :
      ( ( insert_o @ X2 @ ( insert_o @ X2 @ A3 ) )
      = ( insert_o @ X2 @ A3 ) ) ).

% insert_absorb2
thf(fact_2179_insert__absorb2,axiom,
    ! [X2: nat,A3: set_nat] :
      ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A3 ) )
      = ( insert_nat @ X2 @ A3 ) ) ).

% insert_absorb2
thf(fact_2180_insert__absorb2,axiom,
    ! [X2: int,A3: set_int] :
      ( ( insert_int @ X2 @ ( insert_int @ X2 @ A3 ) )
      = ( insert_int @ X2 @ A3 ) ) ).

% insert_absorb2
thf(fact_2181_insert__iff,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member8440522571783428010at_nat @ A2 @ A3 ) ) ) ).

% insert_iff
thf(fact_2182_insert__iff,axiom,
    ! [A2: real,B3: real,A3: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member_real @ A2 @ A3 ) ) ) ).

% insert_iff
thf(fact_2183_insert__iff,axiom,
    ! [A2: $o,B3: $o,A3: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member_o @ A2 @ A3 ) ) ) ).

% insert_iff
thf(fact_2184_insert__iff,axiom,
    ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member_set_nat @ A2 @ A3 ) ) ) ).

% insert_iff
thf(fact_2185_insert__iff,axiom,
    ! [A2: nat,B3: nat,A3: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member_nat @ A2 @ A3 ) ) ) ).

% insert_iff
thf(fact_2186_insert__iff,axiom,
    ! [A2: int,B3: int,A3: set_int] :
      ( ( member_int @ A2 @ ( insert_int @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member_int @ A2 @ A3 ) ) ) ).

% insert_iff
thf(fact_2187_insertCI,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
       => ( A2 = B3 ) )
     => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_2188_insertCI,axiom,
    ! [A2: real,B2: set_real,B3: real] :
      ( ( ~ ( member_real @ A2 @ B2 )
       => ( A2 = B3 ) )
     => ( member_real @ A2 @ ( insert_real @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_2189_insertCI,axiom,
    ! [A2: $o,B2: set_o,B3: $o] :
      ( ( ~ ( member_o @ A2 @ B2 )
       => ( A2 = B3 ) )
     => ( member_o @ A2 @ ( insert_o @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_2190_insertCI,axiom,
    ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
      ( ( ~ ( member_set_nat @ A2 @ B2 )
       => ( A2 = B3 ) )
     => ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_2191_insertCI,axiom,
    ! [A2: nat,B2: set_nat,B3: nat] :
      ( ( ~ ( member_nat @ A2 @ B2 )
       => ( A2 = B3 ) )
     => ( member_nat @ A2 @ ( insert_nat @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_2192_insertCI,axiom,
    ! [A2: int,B2: set_int,B3: int] :
      ( ( ~ ( member_int @ A2 @ B2 )
       => ( A2 = B3 ) )
     => ( member_int @ A2 @ ( insert_int @ B3 @ B2 ) ) ) ).

% insertCI
thf(fact_2193_singletonI,axiom,
    ! [A2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_2194_singletonI,axiom,
    ! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_2195_singletonI,axiom,
    ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_2196_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_2197_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_2198_singletonI,axiom,
    ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_2199_finite__insert,axiom,
    ! [A2: real,A3: set_real] :
      ( ( finite_finite_real @ ( insert_real @ A2 @ A3 ) )
      = ( finite_finite_real @ A3 ) ) ).

% finite_insert
thf(fact_2200_finite__insert,axiom,
    ! [A2: $o,A3: set_o] :
      ( ( finite_finite_o @ ( insert_o @ A2 @ A3 ) )
      = ( finite_finite_o @ A3 ) ) ).

% finite_insert
thf(fact_2201_finite__insert,axiom,
    ! [A2: nat,A3: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) )
      = ( finite_finite_nat @ A3 ) ) ).

% finite_insert
thf(fact_2202_finite__insert,axiom,
    ! [A2: int,A3: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A2 @ A3 ) )
      = ( finite_finite_int @ A3 ) ) ).

% finite_insert
thf(fact_2203_finite__insert,axiom,
    ! [A2: complex,A3: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A3 ) )
      = ( finite3207457112153483333omplex @ A3 ) ) ).

% finite_insert
thf(fact_2204_finite__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) )
      = ( finite6177210948735845034at_nat @ A3 ) ) ).

% finite_insert
thf(fact_2205_finite__insert,axiom,
    ! [A2: extended_enat,A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A2 @ A3 ) )
      = ( finite4001608067531595151d_enat @ A3 ) ) ).

% finite_insert
thf(fact_2206_insert__subset,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
      = ( ( member8440522571783428010at_nat @ X2 @ B2 )
        & ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ).

% insert_subset
thf(fact_2207_insert__subset,axiom,
    ! [X2: real,A3: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
      = ( ( member_real @ X2 @ B2 )
        & ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ).

% insert_subset
thf(fact_2208_insert__subset,axiom,
    ! [X2: $o,A3: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
      = ( ( member_o @ X2 @ B2 )
        & ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ).

% insert_subset
thf(fact_2209_insert__subset,axiom,
    ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
      = ( ( member_set_nat @ X2 @ B2 )
        & ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ).

% insert_subset
thf(fact_2210_insert__subset,axiom,
    ! [X2: nat,A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
      = ( ( member_nat @ X2 @ B2 )
        & ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).

% insert_subset
thf(fact_2211_insert__subset,axiom,
    ! [X2: int,A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
      = ( ( member_int @ X2 @ B2 )
        & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).

% insert_subset
thf(fact_2212_insert__Diff1,axiom,
    ! [X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ B2 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
        = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2213_insert__Diff1,axiom,
    ! [X2: real,B2: set_real,A3: set_real] :
      ( ( member_real @ X2 @ B2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
        = ( minus_minus_set_real @ A3 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2214_insert__Diff1,axiom,
    ! [X2: $o,B2: set_o,A3: set_o] :
      ( ( member_o @ X2 @ B2 )
     => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
        = ( minus_minus_set_o @ A3 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2215_insert__Diff1,axiom,
    ! [X2: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X2 @ B2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
        = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2216_insert__Diff1,axiom,
    ! [X2: int,B2: set_int,A3: set_int] :
      ( ( member_int @ X2 @ B2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
        = ( minus_minus_set_int @ A3 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2217_insert__Diff1,axiom,
    ! [X2: nat,B2: set_nat,A3: set_nat] :
      ( ( member_nat @ X2 @ B2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
        = ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_2218_Diff__insert0,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
     => ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
        = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2219_Diff__insert0,axiom,
    ! [X2: real,A3: set_real,B2: set_real] :
      ( ~ ( member_real @ X2 @ A3 )
     => ( ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
        = ( minus_minus_set_real @ A3 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2220_Diff__insert0,axiom,
    ! [X2: $o,A3: set_o,B2: set_o] :
      ( ~ ( member_o @ X2 @ A3 )
     => ( ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
        = ( minus_minus_set_o @ A3 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2221_Diff__insert0,axiom,
    ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X2 @ A3 )
     => ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
        = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2222_Diff__insert0,axiom,
    ! [X2: int,A3: set_int,B2: set_int] :
      ( ~ ( member_int @ X2 @ A3 )
     => ( ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
        = ( minus_minus_set_int @ A3 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2223_Diff__insert0,axiom,
    ! [X2: nat,A3: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X2 @ A3 )
     => ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
        = ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_2224_singleton__insert__inj__eq,axiom,
    ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A2 @ A3 ) )
      = ( ( A2 = B3 )
        & ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2225_singleton__insert__inj__eq,axiom,
    ! [B3: real,A2: real,A3: set_real] :
      ( ( ( insert_real @ B3 @ bot_bot_set_real )
        = ( insert_real @ A2 @ A3 ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2226_singleton__insert__inj__eq,axiom,
    ! [B3: $o,A2: $o,A3: set_o] :
      ( ( ( insert_o @ B3 @ bot_bot_set_o )
        = ( insert_o @ A2 @ A3 ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2227_singleton__insert__inj__eq,axiom,
    ! [B3: nat,A2: nat,A3: set_nat] :
      ( ( ( insert_nat @ B3 @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A3 ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2228_singleton__insert__inj__eq,axiom,
    ! [B3: int,A2: int,A3: set_int] :
      ( ( ( insert_int @ B3 @ bot_bot_set_int )
        = ( insert_int @ A2 @ A3 ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_2229_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ A3 )
        = ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
      = ( ( A2 = B3 )
        & ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2230_singleton__insert__inj__eq_H,axiom,
    ! [A2: real,A3: set_real,B3: real] :
      ( ( ( insert_real @ A2 @ A3 )
        = ( insert_real @ B3 @ bot_bot_set_real ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2231_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A3: set_o,B3: $o] :
      ( ( ( insert_o @ A2 @ A3 )
        = ( insert_o @ B3 @ bot_bot_set_o ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2232_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A3: set_nat,B3: nat] :
      ( ( ( insert_nat @ A2 @ A3 )
        = ( insert_nat @ B3 @ bot_bot_set_nat ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2233_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A3: set_int,B3: int] :
      ( ( ( insert_int @ A2 @ A3 )
        = ( insert_int @ B3 @ bot_bot_set_int ) )
      = ( ( A2 = B3 )
        & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_2234_insert__Diff__single,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A2 @ A3 ) ) ).

% insert_Diff_single
thf(fact_2235_insert__Diff__single,axiom,
    ! [A2: real,A3: set_real] :
      ( ( insert_real @ A2 @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
      = ( insert_real @ A2 @ A3 ) ) ).

% insert_Diff_single
thf(fact_2236_insert__Diff__single,axiom,
    ! [A2: $o,A3: set_o] :
      ( ( insert_o @ A2 @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
      = ( insert_o @ A2 @ A3 ) ) ).

% insert_Diff_single
thf(fact_2237_insert__Diff__single,axiom,
    ! [A2: int,A3: set_int] :
      ( ( insert_int @ A2 @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
      = ( insert_int @ A2 @ A3 ) ) ).

% insert_Diff_single
thf(fact_2238_insert__Diff__single,axiom,
    ! [A2: nat,A3: set_nat] :
      ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A2 @ A3 ) ) ).

% insert_Diff_single
thf(fact_2239_finite__Diff__insert,axiom,
    ! [A3: set_real,A2: real,B2: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2240_finite__Diff__insert,axiom,
    ! [A3: set_o,A2: $o,B2: set_o] :
      ( ( finite_finite_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) ) )
      = ( finite_finite_o @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2241_finite__Diff__insert,axiom,
    ! [A3: set_int,A2: int,B2: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2242_finite__Diff__insert,axiom,
    ! [A3: set_complex,A2: complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ B2 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2243_finite__Diff__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) )
      = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2244_finite__Diff__insert,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ B2 ) ) )
      = ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2245_finite__Diff__insert,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_2246_card__insert__disjoint,axiom,
    ! [A3: set_real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ~ ( member_real @ X2 @ A3 )
       => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
          = ( suc @ ( finite_card_real @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2247_card__insert__disjoint,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X2 @ A3 )
       => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
          = ( suc @ ( finite_card_o @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2248_card__insert__disjoint,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ~ ( member_list_nat @ X2 @ A3 )
       => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
          = ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2249_card__insert__disjoint,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ~ ( member_set_nat @ X2 @ A3 )
       => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
          = ( suc @ ( finite_card_set_nat @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2250_card__insert__disjoint,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ X2 @ A3 )
       => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
          = ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2251_card__insert__disjoint,axiom,
    ! [A3: set_int,X2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ~ ( member_int @ X2 @ A3 )
       => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
          = ( suc @ ( finite_card_int @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2252_card__insert__disjoint,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ~ ( member_complex @ X2 @ A3 )
       => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
          = ( suc @ ( finite_card_complex @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2253_card__insert__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
          = ( suc @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2254_card__insert__disjoint,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ~ ( member_Extended_enat @ X2 @ A3 )
       => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
          = ( suc @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2255_subset__Compl__singleton,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B3 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_2256_subset__Compl__singleton,axiom,
    ! [A3: set_set_nat,B3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B3 @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B3 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_2257_subset__Compl__singleton,axiom,
    ! [A3: set_real,B3: real] :
      ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ ( insert_real @ B3 @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B3 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_2258_subset__Compl__singleton,axiom,
    ! [A3: set_o,B3: $o] :
      ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ ( insert_o @ B3 @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B3 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_2259_subset__Compl__singleton,axiom,
    ! [A3: set_nat,B3: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B3 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_2260_subset__Compl__singleton,axiom,
    ! [A3: set_int,B3: int] :
      ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( insert_int @ B3 @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B3 @ A3 ) ) ) ).

% subset_Compl_singleton
thf(fact_2261_card__Diff__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2262_card__Diff__insert,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( member_real @ A2 @ A3 )
     => ( ~ ( member_real @ A2 @ B2 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2263_card__Diff__insert,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ A2 @ A3 )
     => ( ~ ( member_o @ A2 @ B2 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2264_card__Diff__insert,axiom,
    ! [A2: complex,A3: set_complex,B2: set_complex] :
      ( ( member_complex @ A2 @ A3 )
     => ( ~ ( member_complex @ A2 @ B2 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2265_card__Diff__insert,axiom,
    ! [A2: list_nat,A3: set_list_nat,B2: set_list_nat] :
      ( ( member_list_nat @ A2 @ A3 )
     => ( ~ ( member_list_nat @ A2 @ B2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2266_card__Diff__insert,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ A2 @ A3 )
     => ( ~ ( member_set_nat @ A2 @ B2 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2267_card__Diff__insert,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( member_int @ A2 @ A3 )
     => ( ~ ( member_int @ A2 @ B2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2268_card__Diff__insert,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ A2 @ A3 )
     => ( ~ ( member_nat @ A2 @ B2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_2269_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_2270_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X4: real] : ( member_real @ X4 @ S )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( member_real @ X5 @ S )
           => ( ord_less_eq_real @ X5 @ Z5 ) )
       => ? [Y4: real] :
            ( ! [X4: real] :
                ( ( member_real @ X4 @ S )
               => ( ord_less_eq_real @ X4 @ Y4 ) )
            & ! [Z5: real] :
                ( ! [X5: real] :
                    ( ( member_real @ X5 @ S )
                   => ( ord_less_eq_real @ X5 @ Z5 ) )
               => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_2271_mk__disjoint__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A3 )
     => ? [B8: set_Pr1261947904930325089at_nat] :
          ( ( A3
            = ( insert8211810215607154385at_nat @ A2 @ B8 ) )
          & ~ ( member8440522571783428010at_nat @ A2 @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2272_mk__disjoint__insert,axiom,
    ! [A2: real,A3: set_real] :
      ( ( member_real @ A2 @ A3 )
     => ? [B8: set_real] :
          ( ( A3
            = ( insert_real @ A2 @ B8 ) )
          & ~ ( member_real @ A2 @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2273_mk__disjoint__insert,axiom,
    ! [A2: $o,A3: set_o] :
      ( ( member_o @ A2 @ A3 )
     => ? [B8: set_o] :
          ( ( A3
            = ( insert_o @ A2 @ B8 ) )
          & ~ ( member_o @ A2 @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2274_mk__disjoint__insert,axiom,
    ! [A2: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ A2 @ A3 )
     => ? [B8: set_set_nat] :
          ( ( A3
            = ( insert_set_nat @ A2 @ B8 ) )
          & ~ ( member_set_nat @ A2 @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2275_mk__disjoint__insert,axiom,
    ! [A2: nat,A3: set_nat] :
      ( ( member_nat @ A2 @ A3 )
     => ? [B8: set_nat] :
          ( ( A3
            = ( insert_nat @ A2 @ B8 ) )
          & ~ ( member_nat @ A2 @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2276_mk__disjoint__insert,axiom,
    ! [A2: int,A3: set_int] :
      ( ( member_int @ A2 @ A3 )
     => ? [B8: set_int] :
          ( ( A3
            = ( insert_int @ A2 @ B8 ) )
          & ~ ( member_int @ A2 @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_2277_insert__commute,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ Y3 @ A3 ) )
      = ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ) ).

% insert_commute
thf(fact_2278_insert__commute,axiom,
    ! [X2: real,Y3: real,A3: set_real] :
      ( ( insert_real @ X2 @ ( insert_real @ Y3 @ A3 ) )
      = ( insert_real @ Y3 @ ( insert_real @ X2 @ A3 ) ) ) ).

% insert_commute
thf(fact_2279_insert__commute,axiom,
    ! [X2: $o,Y3: $o,A3: set_o] :
      ( ( insert_o @ X2 @ ( insert_o @ Y3 @ A3 ) )
      = ( insert_o @ Y3 @ ( insert_o @ X2 @ A3 ) ) ) ).

% insert_commute
thf(fact_2280_insert__commute,axiom,
    ! [X2: nat,Y3: nat,A3: set_nat] :
      ( ( insert_nat @ X2 @ ( insert_nat @ Y3 @ A3 ) )
      = ( insert_nat @ Y3 @ ( insert_nat @ X2 @ A3 ) ) ) ).

% insert_commute
thf(fact_2281_insert__commute,axiom,
    ! [X2: int,Y3: int,A3: set_int] :
      ( ( insert_int @ X2 @ ( insert_int @ Y3 @ A3 ) )
      = ( insert_int @ Y3 @ ( insert_int @ X2 @ A3 ) ) ) ).

% insert_commute
thf(fact_2282_insert__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ B3 @ B2 )
       => ( ( ( insert8211810215607154385at_nat @ A2 @ A3 )
            = ( insert8211810215607154385at_nat @ B3 @ B2 ) )
          = ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_Pr1261947904930325089at_nat] :
                  ( ( A3
                    = ( insert8211810215607154385at_nat @ B3 @ C4 ) )
                  & ~ ( member8440522571783428010at_nat @ B3 @ C4 )
                  & ( B2
                    = ( insert8211810215607154385at_nat @ A2 @ C4 ) )
                  & ~ ( member8440522571783428010at_nat @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2283_insert__eq__iff,axiom,
    ! [A2: real,A3: set_real,B3: real,B2: set_real] :
      ( ~ ( member_real @ A2 @ A3 )
     => ( ~ ( member_real @ B3 @ B2 )
       => ( ( ( insert_real @ A2 @ A3 )
            = ( insert_real @ B3 @ B2 ) )
          = ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_real] :
                  ( ( A3
                    = ( insert_real @ B3 @ C4 ) )
                  & ~ ( member_real @ B3 @ C4 )
                  & ( B2
                    = ( insert_real @ A2 @ C4 ) )
                  & ~ ( member_real @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2284_insert__eq__iff,axiom,
    ! [A2: $o,A3: set_o,B3: $o,B2: set_o] :
      ( ~ ( member_o @ A2 @ A3 )
     => ( ~ ( member_o @ B3 @ B2 )
       => ( ( ( insert_o @ A2 @ A3 )
            = ( insert_o @ B3 @ B2 ) )
          = ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 = ~ B3 )
             => ? [C4: set_o] :
                  ( ( A3
                    = ( insert_o @ B3 @ C4 ) )
                  & ~ ( member_o @ B3 @ C4 )
                  & ( B2
                    = ( insert_o @ A2 @ C4 ) )
                  & ~ ( member_o @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2285_insert__eq__iff,axiom,
    ! [A2: set_nat,A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A2 @ A3 )
     => ( ~ ( member_set_nat @ B3 @ B2 )
       => ( ( ( insert_set_nat @ A2 @ A3 )
            = ( insert_set_nat @ B3 @ B2 ) )
          = ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_set_nat] :
                  ( ( A3
                    = ( insert_set_nat @ B3 @ C4 ) )
                  & ~ ( member_set_nat @ B3 @ C4 )
                  & ( B2
                    = ( insert_set_nat @ A2 @ C4 ) )
                  & ~ ( member_set_nat @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2286_insert__eq__iff,axiom,
    ! [A2: nat,A3: set_nat,B3: nat,B2: set_nat] :
      ( ~ ( member_nat @ A2 @ A3 )
     => ( ~ ( member_nat @ B3 @ B2 )
       => ( ( ( insert_nat @ A2 @ A3 )
            = ( insert_nat @ B3 @ B2 ) )
          = ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_nat] :
                  ( ( A3
                    = ( insert_nat @ B3 @ C4 ) )
                  & ~ ( member_nat @ B3 @ C4 )
                  & ( B2
                    = ( insert_nat @ A2 @ C4 ) )
                  & ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2287_insert__eq__iff,axiom,
    ! [A2: int,A3: set_int,B3: int,B2: set_int] :
      ( ~ ( member_int @ A2 @ A3 )
     => ( ~ ( member_int @ B3 @ B2 )
       => ( ( ( insert_int @ A2 @ A3 )
            = ( insert_int @ B3 @ B2 ) )
          = ( ( ( A2 = B3 )
             => ( A3 = B2 ) )
            & ( ( A2 != B3 )
             => ? [C4: set_int] :
                  ( ( A3
                    = ( insert_int @ B3 @ C4 ) )
                  & ~ ( member_int @ B3 @ C4 )
                  & ( B2
                    = ( insert_int @ A2 @ C4 ) )
                  & ~ ( member_int @ A2 @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_2288_insert__absorb,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A3 )
     => ( ( insert8211810215607154385at_nat @ A2 @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_2289_insert__absorb,axiom,
    ! [A2: real,A3: set_real] :
      ( ( member_real @ A2 @ A3 )
     => ( ( insert_real @ A2 @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_2290_insert__absorb,axiom,
    ! [A2: $o,A3: set_o] :
      ( ( member_o @ A2 @ A3 )
     => ( ( insert_o @ A2 @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_2291_insert__absorb,axiom,
    ! [A2: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ A2 @ A3 )
     => ( ( insert_set_nat @ A2 @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_2292_insert__absorb,axiom,
    ! [A2: nat,A3: set_nat] :
      ( ( member_nat @ A2 @ A3 )
     => ( ( insert_nat @ A2 @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_2293_insert__absorb,axiom,
    ! [A2: int,A3: set_int] :
      ( ( member_int @ A2 @ A3 )
     => ( ( insert_int @ A2 @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_2294_insert__ident,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
       => ( ( ( insert8211810215607154385at_nat @ X2 @ A3 )
            = ( insert8211810215607154385at_nat @ X2 @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2295_insert__ident,axiom,
    ! [X2: real,A3: set_real,B2: set_real] :
      ( ~ ( member_real @ X2 @ A3 )
     => ( ~ ( member_real @ X2 @ B2 )
       => ( ( ( insert_real @ X2 @ A3 )
            = ( insert_real @ X2 @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2296_insert__ident,axiom,
    ! [X2: $o,A3: set_o,B2: set_o] :
      ( ~ ( member_o @ X2 @ A3 )
     => ( ~ ( member_o @ X2 @ B2 )
       => ( ( ( insert_o @ X2 @ A3 )
            = ( insert_o @ X2 @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2297_insert__ident,axiom,
    ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X2 @ A3 )
     => ( ~ ( member_set_nat @ X2 @ B2 )
       => ( ( ( insert_set_nat @ X2 @ A3 )
            = ( insert_set_nat @ X2 @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2298_insert__ident,axiom,
    ! [X2: nat,A3: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X2 @ A3 )
     => ( ~ ( member_nat @ X2 @ B2 )
       => ( ( ( insert_nat @ X2 @ A3 )
            = ( insert_nat @ X2 @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2299_insert__ident,axiom,
    ! [X2: int,A3: set_int,B2: set_int] :
      ( ~ ( member_int @ X2 @ A3 )
     => ( ~ ( member_int @ X2 @ B2 )
       => ( ( ( insert_int @ X2 @ A3 )
            = ( insert_int @ X2 @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% insert_ident
thf(fact_2300_Set_Oset__insert,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ A3 )
     => ~ ! [B8: set_Pr1261947904930325089at_nat] :
            ( ( A3
              = ( insert8211810215607154385at_nat @ X2 @ B8 ) )
           => ( member8440522571783428010at_nat @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2301_Set_Oset__insert,axiom,
    ! [X2: real,A3: set_real] :
      ( ( member_real @ X2 @ A3 )
     => ~ ! [B8: set_real] :
            ( ( A3
              = ( insert_real @ X2 @ B8 ) )
           => ( member_real @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2302_Set_Oset__insert,axiom,
    ! [X2: $o,A3: set_o] :
      ( ( member_o @ X2 @ A3 )
     => ~ ! [B8: set_o] :
            ( ( A3
              = ( insert_o @ X2 @ B8 ) )
           => ( member_o @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2303_Set_Oset__insert,axiom,
    ! [X2: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X2 @ A3 )
     => ~ ! [B8: set_set_nat] :
            ( ( A3
              = ( insert_set_nat @ X2 @ B8 ) )
           => ( member_set_nat @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2304_Set_Oset__insert,axiom,
    ! [X2: nat,A3: set_nat] :
      ( ( member_nat @ X2 @ A3 )
     => ~ ! [B8: set_nat] :
            ( ( A3
              = ( insert_nat @ X2 @ B8 ) )
           => ( member_nat @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2305_Set_Oset__insert,axiom,
    ! [X2: int,A3: set_int] :
      ( ( member_int @ X2 @ A3 )
     => ~ ! [B8: set_int] :
            ( ( A3
              = ( insert_int @ X2 @ B8 ) )
           => ( member_int @ X2 @ B8 ) ) ) ).

% Set.set_insert
thf(fact_2306_insertI2,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ B2 )
     => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_2307_insertI2,axiom,
    ! [A2: real,B2: set_real,B3: real] :
      ( ( member_real @ A2 @ B2 )
     => ( member_real @ A2 @ ( insert_real @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_2308_insertI2,axiom,
    ! [A2: $o,B2: set_o,B3: $o] :
      ( ( member_o @ A2 @ B2 )
     => ( member_o @ A2 @ ( insert_o @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_2309_insertI2,axiom,
    ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
      ( ( member_set_nat @ A2 @ B2 )
     => ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_2310_insertI2,axiom,
    ! [A2: nat,B2: set_nat,B3: nat] :
      ( ( member_nat @ A2 @ B2 )
     => ( member_nat @ A2 @ ( insert_nat @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_2311_insertI2,axiom,
    ! [A2: int,B2: set_int,B3: int] :
      ( ( member_int @ A2 @ B2 )
     => ( member_int @ A2 @ ( insert_int @ B3 @ B2 ) ) ) ).

% insertI2
thf(fact_2312_insertI1,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) ).

% insertI1
thf(fact_2313_insertI1,axiom,
    ! [A2: real,B2: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B2 ) ) ).

% insertI1
thf(fact_2314_insertI1,axiom,
    ! [A2: $o,B2: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B2 ) ) ).

% insertI1
thf(fact_2315_insertI1,axiom,
    ! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).

% insertI1
thf(fact_2316_insertI1,axiom,
    ! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).

% insertI1
thf(fact_2317_insertI1,axiom,
    ! [A2: int,B2: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B2 ) ) ).

% insertI1
thf(fact_2318_insertE,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ A3 ) )
     => ( ( A2 != B3 )
       => ( member8440522571783428010at_nat @ A2 @ A3 ) ) ) ).

% insertE
thf(fact_2319_insertE,axiom,
    ! [A2: real,B3: real,A3: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B3 @ A3 ) )
     => ( ( A2 != B3 )
       => ( member_real @ A2 @ A3 ) ) ) ).

% insertE
thf(fact_2320_insertE,axiom,
    ! [A2: $o,B3: $o,A3: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B3 @ A3 ) )
     => ( ( A2 = ~ B3 )
       => ( member_o @ A2 @ A3 ) ) ) ).

% insertE
thf(fact_2321_insertE,axiom,
    ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ A3 ) )
     => ( ( A2 != B3 )
       => ( member_set_nat @ A2 @ A3 ) ) ) ).

% insertE
thf(fact_2322_insertE,axiom,
    ! [A2: nat,B3: nat,A3: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A3 ) )
     => ( ( A2 != B3 )
       => ( member_nat @ A2 @ A3 ) ) ) ).

% insertE
thf(fact_2323_insertE,axiom,
    ! [A2: int,B3: int,A3: set_int] :
      ( ( member_int @ A2 @ ( insert_int @ B3 @ A3 ) )
     => ( ( A2 != B3 )
       => ( member_int @ A2 @ A3 ) ) ) ).

% insertE
thf(fact_2324_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X2
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X2
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F3: nat > nat > nat,A: nat,B: nat] :
              ( X2
             != ( produc8929957630744042906on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_2325_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X2
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( X2
             != ( produc2899441246263362727at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_2326_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X2
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X2
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F3: num > num > num,A: num,B: num] :
              ( X2
             != ( produc5778274026573060048on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_2327_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X2
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X2
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F3: nat > nat > $o,X5: nat,Y4: nat] :
              ( X2
             != ( produc4035269172776083154on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_2328_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X2
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
              ( X2
             != ( produc3994169339658061776at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_2329_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X2
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X2
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F3: num > num > $o,X5: num,Y4: num] :
              ( X2
             != ( produc3576312749637752826on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_2330_singletonD,axiom,
    ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
     => ( B3 = A2 ) ) ).

% singletonD
thf(fact_2331_singletonD,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
     => ( B3 = A2 ) ) ).

% singletonD
thf(fact_2332_singletonD,axiom,
    ! [B3: real,A2: real] :
      ( ( member_real @ B3 @ ( insert_real @ A2 @ bot_bot_set_real ) )
     => ( B3 = A2 ) ) ).

% singletonD
thf(fact_2333_singletonD,axiom,
    ! [B3: $o,A2: $o] :
      ( ( member_o @ B3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B3 = A2 ) ) ).

% singletonD
thf(fact_2334_singletonD,axiom,
    ! [B3: nat,A2: nat] :
      ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B3 = A2 ) ) ).

% singletonD
thf(fact_2335_singletonD,axiom,
    ! [B3: int,A2: int] :
      ( ( member_int @ B3 @ ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( B3 = A2 ) ) ).

% singletonD
thf(fact_2336_singleton__iff,axiom,
    ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
      = ( B3 = A2 ) ) ).

% singleton_iff
thf(fact_2337_singleton__iff,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
      = ( B3 = A2 ) ) ).

% singleton_iff
thf(fact_2338_singleton__iff,axiom,
    ! [B3: real,A2: real] :
      ( ( member_real @ B3 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( B3 = A2 ) ) ).

% singleton_iff
thf(fact_2339_singleton__iff,axiom,
    ! [B3: $o,A2: $o] :
      ( ( member_o @ B3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B3 = A2 ) ) ).

% singleton_iff
thf(fact_2340_singleton__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B3 = A2 ) ) ).

% singleton_iff
thf(fact_2341_singleton__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( member_int @ B3 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( B3 = A2 ) ) ).

% singleton_iff
thf(fact_2342_doubleton__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2343_doubleton__eq__iff,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ( insert_real @ A2 @ ( insert_real @ B3 @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2344_doubleton__eq__iff,axiom,
    ! [A2: $o,B3: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B3 @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2345_doubleton__eq__iff,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2346_doubleton__eq__iff,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ( insert_int @ A2 @ ( insert_int @ B3 @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A2 = C )
          & ( B3 = D ) )
        | ( ( A2 = D )
          & ( B3 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2347_insert__not__empty,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A2 @ A3 )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_2348_insert__not__empty,axiom,
    ! [A2: real,A3: set_real] :
      ( ( insert_real @ A2 @ A3 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_2349_insert__not__empty,axiom,
    ! [A2: $o,A3: set_o] :
      ( ( insert_o @ A2 @ A3 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_2350_insert__not__empty,axiom,
    ! [A2: nat,A3: set_nat] :
      ( ( insert_nat @ A2 @ A3 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_2351_insert__not__empty,axiom,
    ! [A2: int,A3: set_int] :
      ( ( insert_int @ A2 @ A3 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_2352_singleton__inject,axiom,
    ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
     => ( A2 = B3 ) ) ).

% singleton_inject
thf(fact_2353_singleton__inject,axiom,
    ! [A2: real,B3: real] :
      ( ( ( insert_real @ A2 @ bot_bot_set_real )
        = ( insert_real @ B3 @ bot_bot_set_real ) )
     => ( A2 = B3 ) ) ).

% singleton_inject
thf(fact_2354_singleton__inject,axiom,
    ! [A2: $o,B3: $o] :
      ( ( ( insert_o @ A2 @ bot_bot_set_o )
        = ( insert_o @ B3 @ bot_bot_set_o ) )
     => ( A2 = B3 ) ) ).

% singleton_inject
thf(fact_2355_singleton__inject,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B3 @ bot_bot_set_nat ) )
     => ( A2 = B3 ) ) ).

% singleton_inject
thf(fact_2356_singleton__inject,axiom,
    ! [A2: int,B3: int] :
      ( ( ( insert_int @ A2 @ bot_bot_set_int )
        = ( insert_int @ B3 @ bot_bot_set_int ) )
     => ( A2 = B3 ) ) ).

% singleton_inject
thf(fact_2357_finite_OinsertI,axiom,
    ! [A3: set_real,A2: real] :
      ( ( finite_finite_real @ A3 )
     => ( finite_finite_real @ ( insert_real @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2358_finite_OinsertI,axiom,
    ! [A3: set_o,A2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_o @ ( insert_o @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2359_finite_OinsertI,axiom,
    ! [A3: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2360_finite_OinsertI,axiom,
    ! [A3: set_int,A2: int] :
      ( ( finite_finite_int @ A3 )
     => ( finite_finite_int @ ( insert_int @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2361_finite_OinsertI,axiom,
    ! [A3: set_complex,A2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2362_finite_OinsertI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2363_finite_OinsertI,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A2 @ A3 ) ) ) ).

% finite.insertI
thf(fact_2364_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ D4 )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A2 @ C2 ) @ ( insert8211810215607154385at_nat @ A2 @ D4 ) ) ) ).

% insert_mono
thf(fact_2365_insert__mono,axiom,
    ! [C2: set_real,D4: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ C2 @ D4 )
     => ( ord_less_eq_set_real @ ( insert_real @ A2 @ C2 ) @ ( insert_real @ A2 @ D4 ) ) ) ).

% insert_mono
thf(fact_2366_insert__mono,axiom,
    ! [C2: set_o,D4: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ C2 @ D4 )
     => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C2 ) @ ( insert_o @ A2 @ D4 ) ) ) ).

% insert_mono
thf(fact_2367_insert__mono,axiom,
    ! [C2: set_nat,D4: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D4 ) ) ) ).

% insert_mono
thf(fact_2368_insert__mono,axiom,
    ! [C2: set_int,D4: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ C2 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C2 ) @ ( insert_int @ A2 @ D4 ) ) ) ).

% insert_mono
thf(fact_2369_subset__insert,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
        = ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ).

% subset_insert
thf(fact_2370_subset__insert,axiom,
    ! [X2: real,A3: set_real,B2: set_real] :
      ( ~ ( member_real @ X2 @ A3 )
     => ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
        = ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ).

% subset_insert
thf(fact_2371_subset__insert,axiom,
    ! [X2: $o,A3: set_o,B2: set_o] :
      ( ~ ( member_o @ X2 @ A3 )
     => ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
        = ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ).

% subset_insert
thf(fact_2372_subset__insert,axiom,
    ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X2 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
        = ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ).

% subset_insert
thf(fact_2373_subset__insert,axiom,
    ! [X2: nat,A3: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X2 @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
        = ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).

% subset_insert
thf(fact_2374_subset__insert,axiom,
    ! [X2: int,A3: set_int,B2: set_int] :
      ( ~ ( member_int @ X2 @ A3 )
     => ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
        = ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).

% subset_insert
thf(fact_2375_subset__insertI,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_2376_subset__insertI,axiom,
    ! [B2: set_real,A2: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_2377_subset__insertI,axiom,
    ! [B2: set_o,A2: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_2378_subset__insertI,axiom,
    ! [B2: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_2379_subset__insertI,axiom,
    ! [B2: set_int,A2: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A2 @ B2 ) ) ).

% subset_insertI
thf(fact_2380_subset__insertI2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
     => ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2381_subset__insertI2,axiom,
    ! [A3: set_real,B2: set_real,B3: real] :
      ( ( ord_less_eq_set_real @ A3 @ B2 )
     => ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2382_subset__insertI2,axiom,
    ! [A3: set_o,B2: set_o,B3: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B2 )
     => ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2383_subset__insertI2,axiom,
    ! [A3: set_nat,B2: set_nat,B3: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2384_subset__insertI2,axiom,
    ! [A3: set_int,B2: set_int,B3: int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ B2 ) ) ) ).

% subset_insertI2
thf(fact_2385_insert__subsetI,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ X6 @ A3 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ X6 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_2386_insert__subsetI,axiom,
    ! [X2: real,A3: set_real,X6: set_real] :
      ( ( member_real @ X2 @ A3 )
     => ( ( ord_less_eq_set_real @ X6 @ A3 )
       => ( ord_less_eq_set_real @ ( insert_real @ X2 @ X6 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_2387_insert__subsetI,axiom,
    ! [X2: $o,A3: set_o,X6: set_o] :
      ( ( member_o @ X2 @ A3 )
     => ( ( ord_less_eq_set_o @ X6 @ A3 )
       => ( ord_less_eq_set_o @ ( insert_o @ X2 @ X6 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_2388_insert__subsetI,axiom,
    ! [X2: set_nat,A3: set_set_nat,X6: set_set_nat] :
      ( ( member_set_nat @ X2 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ X6 @ A3 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ X6 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_2389_insert__subsetI,axiom,
    ! [X2: nat,A3: set_nat,X6: set_nat] :
      ( ( member_nat @ X2 @ A3 )
     => ( ( ord_less_eq_set_nat @ X6 @ A3 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X6 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_2390_insert__subsetI,axiom,
    ! [X2: int,A3: set_int,X6: set_int] :
      ( ( member_int @ X2 @ A3 )
     => ( ( ord_less_eq_set_int @ X6 @ A3 )
       => ( ord_less_eq_set_int @ ( insert_int @ X2 @ X6 ) @ A3 ) ) ) ).

% insert_subsetI
thf(fact_2391_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
          ( X2
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_2392_insert__Diff__if,axiom,
    ! [X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X2 @ B2 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
          = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
          = ( insert8211810215607154385at_nat @ X2 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2393_insert__Diff__if,axiom,
    ! [X2: real,B2: set_real,A3: set_real] :
      ( ( ( member_real @ X2 @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
          = ( minus_minus_set_real @ A3 @ B2 ) ) )
      & ( ~ ( member_real @ X2 @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
          = ( insert_real @ X2 @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2394_insert__Diff__if,axiom,
    ! [X2: $o,B2: set_o,A3: set_o] :
      ( ( ( member_o @ X2 @ B2 )
       => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
          = ( minus_minus_set_o @ A3 @ B2 ) ) )
      & ( ~ ( member_o @ X2 @ B2 )
       => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
          = ( insert_o @ X2 @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2395_insert__Diff__if,axiom,
    ! [X2: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( ( ( member_set_nat @ X2 @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
          = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) )
      & ( ~ ( member_set_nat @ X2 @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
          = ( insert_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2396_insert__Diff__if,axiom,
    ! [X2: int,B2: set_int,A3: set_int] :
      ( ( ( member_int @ X2 @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
          = ( minus_minus_set_int @ A3 @ B2 ) ) )
      & ( ~ ( member_int @ X2 @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
          = ( insert_int @ X2 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2397_insert__Diff__if,axiom,
    ! [X2: nat,B2: set_nat,A3: set_nat] :
      ( ( ( member_nat @ X2 @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
          = ( minus_minus_set_nat @ A3 @ B2 ) ) )
      & ( ~ ( member_nat @ X2 @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
          = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2398_finite_Ocases,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ~ ! [A5: set_complex] :
              ( ? [A: complex] :
                  ( A2
                  = ( insert_complex @ A @ A5 ) )
             => ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2399_finite_Ocases,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( A2 != bot_bo2099793752762293965at_nat )
       => ~ ! [A5: set_Pr1261947904930325089at_nat] :
              ( ? [A: product_prod_nat_nat] :
                  ( A2
                  = ( insert8211810215607154385at_nat @ A @ A5 ) )
             => ~ ( finite6177210948735845034at_nat @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2400_finite_Ocases,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ~ ! [A5: set_Extended_enat] :
              ( ? [A: extended_enat] :
                  ( A2
                  = ( insert_Extended_enat @ A @ A5 ) )
             => ~ ( finite4001608067531595151d_enat @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2401_finite_Ocases,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ~ ! [A5: set_real] :
              ( ? [A: real] :
                  ( A2
                  = ( insert_real @ A @ A5 ) )
             => ~ ( finite_finite_real @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2402_finite_Ocases,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ~ ! [A5: set_o] :
              ( ? [A: $o] :
                  ( A2
                  = ( insert_o @ A @ A5 ) )
             => ~ ( finite_finite_o @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2403_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A5: set_nat] :
              ( ? [A: nat] :
                  ( A2
                  = ( insert_nat @ A @ A5 ) )
             => ~ ( finite_finite_nat @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2404_finite_Ocases,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ~ ! [A5: set_int] :
              ( ? [A: int] :
                  ( A2
                  = ( insert_int @ A @ A5 ) )
             => ~ ( finite_finite_int @ A5 ) ) ) ) ).

% finite.cases
thf(fact_2405_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A4: set_complex] :
          ( ( A4 = bot_bot_set_complex )
          | ? [A6: set_complex,B4: complex] :
              ( ( A4
                = ( insert_complex @ B4 @ A6 ) )
              & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2406_finite_Osimps,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] :
          ( ( A4 = bot_bo2099793752762293965at_nat )
          | ? [A6: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
              ( ( A4
                = ( insert8211810215607154385at_nat @ B4 @ A6 ) )
              & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2407_finite_Osimps,axiom,
    ( finite4001608067531595151d_enat
    = ( ^ [A4: set_Extended_enat] :
          ( ( A4 = bot_bo7653980558646680370d_enat )
          | ? [A6: set_Extended_enat,B4: extended_enat] :
              ( ( A4
                = ( insert_Extended_enat @ B4 @ A6 ) )
              & ( finite4001608067531595151d_enat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2408_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A4: set_real] :
          ( ( A4 = bot_bot_set_real )
          | ? [A6: set_real,B4: real] :
              ( ( A4
                = ( insert_real @ B4 @ A6 ) )
              & ( finite_finite_real @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2409_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A4: set_o] :
          ( ( A4 = bot_bot_set_o )
          | ? [A6: set_o,B4: $o] :
              ( ( A4
                = ( insert_o @ B4 @ A6 ) )
              & ( finite_finite_o @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2410_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
          ( ( A4 = bot_bot_set_nat )
          | ? [A6: set_nat,B4: nat] :
              ( ( A4
                = ( insert_nat @ B4 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2411_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A4: set_int] :
          ( ( A4 = bot_bot_set_int )
          | ? [A6: set_int,B4: int] :
              ( ( A4
                = ( insert_int @ B4 @ A6 ) )
              & ( finite_finite_int @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_2412_finite__induct,axiom,
    ! [F2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2413_finite__induct,axiom,
    ! [F2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2414_finite__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F4 )
             => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2415_finite__induct,axiom,
    ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F4 )
             => ( ~ ( member_Extended_enat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2416_finite__induct,axiom,
    ! [F2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2417_finite__induct,axiom,
    ! [F2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2418_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2419_finite__induct,axiom,
    ! [F2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2420_finite__ne__induct,axiom,
    ! [F2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( F2 != bot_bot_set_set_nat )
       => ( ! [X5: set_nat] : ( P @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
         => ( ! [X5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2421_finite__ne__induct,axiom,
    ! [F2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( F2 != bot_bot_set_complex )
       => ( ! [X5: complex] : ( P @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
         => ( ! [X5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2422_finite__ne__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( F2 != bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) )
         => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F4 )
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2423_finite__ne__induct,axiom,
    ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( F2 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] : ( P @ ( insert_Extended_enat @ X5 @ bot_bo7653980558646680370d_enat ) )
         => ( ! [X5: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( F4 != bot_bo7653980558646680370d_enat )
                 => ( ~ ( member_Extended_enat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2424_finite__ne__induct,axiom,
    ! [F2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( F2 != bot_bot_set_real )
       => ( ! [X5: real] : ( P @ ( insert_real @ X5 @ bot_bot_set_real ) )
         => ( ! [X5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ ( member_real @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2425_finite__ne__induct,axiom,
    ! [F2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X5: $o] : ( P @ ( insert_o @ X5 @ bot_bot_set_o ) )
         => ( ! [X5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ ( member_o @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2426_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
         => ( ! [X5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2427_finite__ne__induct,axiom,
    ! [F2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( F2 != bot_bot_set_int )
       => ( ! [X5: int] : ( P @ ( insert_int @ X5 @ bot_bot_set_int ) )
         => ( ! [X5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ ( member_int @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2428_infinite__finite__induct,axiom,
    ! [P: set_set_nat > $o,A3: set_set_nat] :
      ( ! [A5: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2429_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A3: set_complex] :
      ( ! [A5: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2430_infinite__finite__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,A3: set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat] :
          ( ~ ( finite6177210948735845034at_nat @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F4 )
             => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2431_infinite__finite__induct,axiom,
    ! [P: set_Extended_enat > $o,A3: set_Extended_enat] :
      ( ! [A5: set_Extended_enat] :
          ( ~ ( finite4001608067531595151d_enat @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F4 )
             => ( ~ ( member_Extended_enat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2432_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A3: set_real] :
      ( ! [A5: set_real] :
          ( ~ ( finite_finite_real @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2433_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A3: set_o] :
      ( ! [A5: set_o] :
          ( ~ ( finite_finite_o @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2434_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A3: set_nat] :
      ( ! [A5: set_nat] :
          ( ~ ( finite_finite_nat @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2435_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A3: set_int] :
      ( ! [A5: set_int] :
          ( ~ ( finite_finite_int @ A5 )
         => ( P @ A5 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2436_subset__singletonD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
     => ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( A3
          = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_2437_subset__singletonD,axiom,
    ! [A3: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) )
     => ( ( A3 = bot_bot_set_real )
        | ( A3
          = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_2438_subset__singletonD,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) )
     => ( ( A3 = bot_bot_set_o )
        | ( A3
          = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_2439_subset__singletonD,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
     => ( ( A3 = bot_bot_set_nat )
        | ( A3
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_2440_subset__singletonD,axiom,
    ! [A3: set_int,X2: int] :
      ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) )
     => ( ( A3 = bot_bot_set_int )
        | ( A3
          = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_2441_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X6 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_2442_subset__singleton__iff,axiom,
    ! [X6: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ X6 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( ( X6 = bot_bot_set_real )
        | ( X6
          = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_2443_subset__singleton__iff,axiom,
    ! [X6: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X6 = bot_bot_set_o )
        | ( X6
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_2444_subset__singleton__iff,axiom,
    ! [X6: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X6 = bot_bot_set_nat )
        | ( X6
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_2445_subset__singleton__iff,axiom,
    ! [X6: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ X6 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( ( X6 = bot_bot_set_int )
        | ( X6
          = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_2446_Diff__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_2447_Diff__insert,axiom,
    ! [A3: set_real,A2: real,B2: set_real] :
      ( ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ B2 ) @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_2448_Diff__insert,axiom,
    ! [A3: set_o,A2: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ B2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_2449_Diff__insert,axiom,
    ! [A3: set_int,A2: int,B2: set_int] :
      ( ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_2450_Diff__insert,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_2451_insert__Diff,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A3 )
     => ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2452_insert__Diff,axiom,
    ! [A2: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ A2 @ A3 )
     => ( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2453_insert__Diff,axiom,
    ! [A2: real,A3: set_real] :
      ( ( member_real @ A2 @ A3 )
     => ( ( insert_real @ A2 @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2454_insert__Diff,axiom,
    ! [A2: $o,A3: set_o] :
      ( ( member_o @ A2 @ A3 )
     => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2455_insert__Diff,axiom,
    ! [A2: int,A3: set_int] :
      ( ( member_int @ A2 @ A3 )
     => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2456_insert__Diff,axiom,
    ! [A2: nat,A3: set_nat] :
      ( ( member_nat @ A2 @ A3 )
     => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2457_Diff__insert2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_2458_Diff__insert2,axiom,
    ! [A3: set_real,A2: real,B2: set_real] :
      ( ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_2459_Diff__insert2,axiom,
    ! [A3: set_o,A2: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_2460_Diff__insert2,axiom,
    ! [A3: set_int,A2: int,B2: set_int] :
      ( ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_2461_Diff__insert2,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_2462_Diff__insert__absorb,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2463_Diff__insert__absorb,axiom,
    ! [X2: set_nat,A3: set_set_nat] :
      ( ~ ( member_set_nat @ X2 @ A3 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2464_Diff__insert__absorb,axiom,
    ! [X2: real,A3: set_real] :
      ( ~ ( member_real @ X2 @ A3 )
     => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2465_Diff__insert__absorb,axiom,
    ! [X2: $o,A3: set_o] :
      ( ~ ( member_o @ X2 @ A3 )
     => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2466_Diff__insert__absorb,axiom,
    ! [X2: int,A3: set_int] :
      ( ~ ( member_int @ X2 @ A3 )
     => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2467_Diff__insert__absorb,axiom,
    ! [X2: nat,A3: set_nat] :
      ( ~ ( member_nat @ X2 @ A3 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2468_subset__Diff__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X2 @ C2 ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) )
        & ~ ( member8440522571783428010at_nat @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2469_subset__Diff__insert,axiom,
    ! [A3: set_real,B2: set_real,X2: real,C2: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B2 @ C2 ) )
        & ~ ( member_real @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2470_subset__Diff__insert,axiom,
    ! [A3: set_o,B2: set_o,X2: $o,C2: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B2 @ C2 ) )
        & ~ ( member_o @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2471_subset__Diff__insert,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,X2: set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X2 @ C2 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
        & ~ ( member_set_nat @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2472_subset__Diff__insert,axiom,
    ! [A3: set_nat,B2: set_nat,X2: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ C2 ) )
        & ~ ( member_nat @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2473_subset__Diff__insert,axiom,
    ! [A3: set_int,B2: set_int,X2: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X2 @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B2 @ C2 ) )
        & ~ ( member_int @ X2 @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2474_card__insert__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2475_card__insert__le,axiom,
    ! [A3: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ ( insert_real @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2476_card__insert__le,axiom,
    ! [A3: set_o,X2: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ ( insert_o @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2477_card__insert__le,axiom,
    ! [A3: set_complex,X2: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2478_card__insert__le,axiom,
    ! [A3: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2479_card__insert__le,axiom,
    ! [A3: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2480_card__insert__le,axiom,
    ! [A3: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2481_card__insert__le,axiom,
    ! [A3: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( insert_int @ X2 @ A3 ) ) ) ).

% card_insert_le
thf(fact_2482_finite__ranking__induct,axiom,
    ! [S: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2483_finite__ranking__induct,axiom,
    ! [S: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,S4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S4 )
             => ( ! [Y5: extended_enat] :
                    ( ( member_Extended_enat @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2484_finite__ranking__induct,axiom,
    ! [S: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2485_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2486_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2487_finite__ranking__induct,axiom,
    ! [S: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2488_finite__ranking__induct,axiom,
    ! [S: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2489_finite__ranking__induct,axiom,
    ! [S: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > num] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,S4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S4 )
             => ( ! [Y5: extended_enat] :
                    ( ( member_Extended_enat @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2490_finite__ranking__induct,axiom,
    ! [S: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2491_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > num] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2492_finite__linorder__max__induct,axiom,
    ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B: extended_enat,A5: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A5 )
             => ( ! [X4: extended_enat] :
                    ( ( member_Extended_enat @ X4 @ A5 )
                   => ( ord_le72135733267957522d_enat @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_Extended_enat @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2493_finite__linorder__max__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B: $o,A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ! [X4: $o] :
                    ( ( member_o @ X4 @ A5 )
                   => ( ord_less_o @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_o @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2494_finite__linorder__max__induct,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B: real,A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A5 )
                   => ( ord_less_real @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_real @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2495_finite__linorder__max__induct,axiom,
    ! [A3: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A3 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B: rat,A5: set_rat] :
              ( ( finite_finite_rat @ A5 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A5 )
                   => ( ord_less_rat @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_rat @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2496_finite__linorder__max__induct,axiom,
    ! [A3: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A3 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B: num,A5: set_num] :
              ( ( finite_finite_num @ A5 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A5 )
                   => ( ord_less_num @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_num @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2497_finite__linorder__max__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B: nat,A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A5 )
                   => ( ord_less_nat @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_nat @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2498_finite__linorder__max__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B: int,A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A5 )
                   => ( ord_less_int @ X4 @ B ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_int @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2499_finite__linorder__min__induct,axiom,
    ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B: extended_enat,A5: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A5 )
             => ( ! [X4: extended_enat] :
                    ( ( member_Extended_enat @ X4 @ A5 )
                   => ( ord_le72135733267957522d_enat @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_Extended_enat @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2500_finite__linorder__min__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B: $o,A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ! [X4: $o] :
                    ( ( member_o @ X4 @ A5 )
                   => ( ord_less_o @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_o @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2501_finite__linorder__min__induct,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B: real,A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A5 )
                   => ( ord_less_real @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_real @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2502_finite__linorder__min__induct,axiom,
    ! [A3: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A3 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B: rat,A5: set_rat] :
              ( ( finite_finite_rat @ A5 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A5 )
                   => ( ord_less_rat @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_rat @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2503_finite__linorder__min__induct,axiom,
    ! [A3: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A3 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B: num,A5: set_num] :
              ( ( finite_finite_num @ A5 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A5 )
                   => ( ord_less_num @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_num @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2504_finite__linorder__min__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B: nat,A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A5 )
                   => ( ord_less_nat @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_nat @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2505_finite__linorder__min__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B: int,A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A5 )
                   => ( ord_less_int @ B @ X4 ) )
               => ( ( P @ A5 )
                 => ( P @ ( insert_int @ B @ A5 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2506_finite__subset__induct,axiom,
    ! [F2: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( ord_le6893508408891458716et_nat @ F2 @ A3 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A @ A3 )
                 => ( ~ ( member_set_nat @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2507_finite__subset__induct,axiom,
    ! [F2: set_complex,A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ord_le211207098394363844omplex @ F2 @ A3 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A @ A3 )
                 => ( ~ ( member_complex @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2508_finite__subset__induct,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( ord_le3146513528884898305at_nat @ F2 @ A3 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F4 )
               => ( ( member8440522571783428010at_nat @ A @ A3 )
                 => ( ~ ( member8440522571783428010at_nat @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert8211810215607154385at_nat @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2509_finite__subset__induct,axiom,
    ! [F2: set_Extended_enat,A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( ord_le7203529160286727270d_enat @ F2 @ A3 )
       => ( ( P @ bot_bo7653980558646680370d_enat )
         => ( ! [A: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( member_Extended_enat @ A @ A3 )
                 => ( ~ ( member_Extended_enat @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_Extended_enat @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2510_finite__subset__induct,axiom,
    ! [F2: set_real,A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( ord_less_eq_set_real @ F2 @ A3 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A @ A3 )
                 => ( ~ ( member_real @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2511_finite__subset__induct,axiom,
    ! [F2: set_o,A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A3 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o @ A @ A3 )
                 => ( ~ ( member_o @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_o @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2512_finite__subset__induct,axiom,
    ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A3 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A @ A3 )
                 => ( ~ ( member_nat @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2513_finite__subset__induct,axiom,
    ! [F2: set_int,A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( ord_less_eq_set_int @ F2 @ A3 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A @ A3 )
                 => ( ~ ( member_int @ A @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ A @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2514_finite__subset__induct_H,axiom,
    ! [F2: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F2 )
     => ( ( ord_le6893508408891458716et_nat @ F2 @ A3 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A @ A3 )
                 => ( ( ord_le6893508408891458716et_nat @ F4 @ A3 )
                   => ( ~ ( member_set_nat @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_set_nat @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2515_finite__subset__induct_H,axiom,
    ! [F2: set_complex,A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ord_le211207098394363844omplex @ F2 @ A3 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A @ A3 )
                 => ( ( ord_le211207098394363844omplex @ F4 @ A3 )
                   => ( ~ ( member_complex @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_complex @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2516_finite__subset__induct_H,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( ord_le3146513528884898305at_nat @ F2 @ A3 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F4 )
               => ( ( member8440522571783428010at_nat @ A @ A3 )
                 => ( ( ord_le3146513528884898305at_nat @ F4 @ A3 )
                   => ( ~ ( member8440522571783428010at_nat @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert8211810215607154385at_nat @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2517_finite__subset__induct_H,axiom,
    ! [F2: set_Extended_enat,A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( ord_le7203529160286727270d_enat @ F2 @ A3 )
       => ( ( P @ bot_bo7653980558646680370d_enat )
         => ( ! [A: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( member_Extended_enat @ A @ A3 )
                 => ( ( ord_le7203529160286727270d_enat @ F4 @ A3 )
                   => ( ~ ( member_Extended_enat @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_Extended_enat @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2518_finite__subset__induct_H,axiom,
    ! [F2: set_real,A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F2 )
     => ( ( ord_less_eq_set_real @ F2 @ A3 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A @ A3 )
                 => ( ( ord_less_eq_set_real @ F4 @ A3 )
                   => ( ~ ( member_real @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_real @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2519_finite__subset__induct_H,axiom,
    ! [F2: set_o,A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A3 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o @ A @ A3 )
                 => ( ( ord_less_eq_set_o @ F4 @ A3 )
                   => ( ~ ( member_o @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_o @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2520_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A3 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A @ A3 )
                 => ( ( ord_less_eq_set_nat @ F4 @ A3 )
                   => ( ~ ( member_nat @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_nat @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2521_finite__subset__induct_H,axiom,
    ! [F2: set_int,A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F2 )
     => ( ( ord_less_eq_set_int @ F2 @ A3 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A @ A3 )
                 => ( ( ord_less_eq_set_int @ F4 @ A3 )
                   => ( ~ ( member_int @ A @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_int @ A @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2522_card__Suc__eq__finite,axiom,
    ! [A3: set_real,K: nat] :
      ( ( ( finite_card_real @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: real,B6: set_real] :
            ( ( A3
              = ( insert_real @ B4 @ B6 ) )
            & ~ ( member_real @ B4 @ B6 )
            & ( ( finite_card_real @ B6 )
              = K )
            & ( finite_finite_real @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2523_card__Suc__eq__finite,axiom,
    ! [A3: set_o,K: nat] :
      ( ( ( finite_card_o @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: $o,B6: set_o] :
            ( ( A3
              = ( insert_o @ B4 @ B6 ) )
            & ~ ( member_o @ B4 @ B6 )
            & ( ( finite_card_o @ B6 )
              = K )
            & ( finite_finite_o @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2524_card__Suc__eq__finite,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: list_nat,B6: set_list_nat] :
            ( ( A3
              = ( insert_list_nat @ B4 @ B6 ) )
            & ~ ( member_list_nat @ B4 @ B6 )
            & ( ( finite_card_list_nat @ B6 )
              = K )
            & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2525_card__Suc__eq__finite,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: set_nat,B6: set_set_nat] :
            ( ( A3
              = ( insert_set_nat @ B4 @ B6 ) )
            & ~ ( member_set_nat @ B4 @ B6 )
            & ( ( finite_card_set_nat @ B6 )
              = K )
            & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2526_card__Suc__eq__finite,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: nat,B6: set_nat] :
            ( ( A3
              = ( insert_nat @ B4 @ B6 ) )
            & ~ ( member_nat @ B4 @ B6 )
            & ( ( finite_card_nat @ B6 )
              = K )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2527_card__Suc__eq__finite,axiom,
    ! [A3: set_int,K: nat] :
      ( ( ( finite_card_int @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: int,B6: set_int] :
            ( ( A3
              = ( insert_int @ B4 @ B6 ) )
            & ~ ( member_int @ B4 @ B6 )
            & ( ( finite_card_int @ B6 )
              = K )
            & ( finite_finite_int @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2528_card__Suc__eq__finite,axiom,
    ! [A3: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: complex,B6: set_complex] :
            ( ( A3
              = ( insert_complex @ B4 @ B6 ) )
            & ~ ( member_complex @ B4 @ B6 )
            & ( ( finite_card_complex @ B6 )
              = K )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2529_card__Suc__eq__finite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A3
              = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
            & ( ( finite711546835091564841at_nat @ B6 )
              = K )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2530_card__Suc__eq__finite,axiom,
    ! [A3: set_Extended_enat,K: nat] :
      ( ( ( finite121521170596916366d_enat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: extended_enat,B6: set_Extended_enat] :
            ( ( A3
              = ( insert_Extended_enat @ B4 @ B6 ) )
            & ~ ( member_Extended_enat @ B4 @ B6 )
            & ( ( finite121521170596916366d_enat @ B6 )
              = K )
            & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2531_card__insert__if,axiom,
    ! [A3: set_real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( member_real @ X2 @ A3 )
         => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
            = ( finite_card_real @ A3 ) ) )
        & ( ~ ( member_real @ X2 @ A3 )
         => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
            = ( suc @ ( finite_card_real @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2532_card__insert__if,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( member_o @ X2 @ A3 )
         => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
            = ( finite_card_o @ A3 ) ) )
        & ( ~ ( member_o @ X2 @ A3 )
         => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
            = ( suc @ ( finite_card_o @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2533_card__insert__if,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( ( member_list_nat @ X2 @ A3 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
            = ( finite_card_list_nat @ A3 ) ) )
        & ( ~ ( member_list_nat @ X2 @ A3 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
            = ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2534_card__insert__if,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( ( member_set_nat @ X2 @ A3 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
            = ( finite_card_set_nat @ A3 ) ) )
        & ( ~ ( member_set_nat @ X2 @ A3 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
            = ( suc @ ( finite_card_set_nat @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2535_card__insert__if,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( member_nat @ X2 @ A3 )
         => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
            = ( finite_card_nat @ A3 ) ) )
        & ( ~ ( member_nat @ X2 @ A3 )
         => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
            = ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2536_card__insert__if,axiom,
    ! [A3: set_int,X2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( member_int @ X2 @ A3 )
         => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
            = ( finite_card_int @ A3 ) ) )
        & ( ~ ( member_int @ X2 @ A3 )
         => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
            = ( suc @ ( finite_card_int @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2537_card__insert__if,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( member_complex @ X2 @ A3 )
         => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
            = ( finite_card_complex @ A3 ) ) )
        & ( ~ ( member_complex @ X2 @ A3 )
         => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
            = ( suc @ ( finite_card_complex @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2538_card__insert__if,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
         => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
            = ( finite711546835091564841at_nat @ A3 ) ) )
        & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
         => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
            = ( suc @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2539_card__insert__if,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( member_Extended_enat @ X2 @ A3 )
         => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
            = ( finite121521170596916366d_enat @ A3 ) ) )
        & ( ~ ( member_Extended_enat @ X2 @ A3 )
         => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
            = ( suc @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2540_infinite__remove,axiom,
    ! [S: set_complex,A2: complex] :
      ( ~ ( finite3207457112153483333omplex @ S )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_2541_infinite__remove,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% infinite_remove
thf(fact_2542_infinite__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S )
     => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% infinite_remove
thf(fact_2543_infinite__remove,axiom,
    ! [S: set_real,A2: real] :
      ( ~ ( finite_finite_real @ S )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_2544_infinite__remove,axiom,
    ! [S: set_o,A2: $o] :
      ( ~ ( finite_finite_o @ S )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_2545_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ ( finite_finite_int @ S )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_2546_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_2547_infinite__coinduct,axiom,
    ! [X6: set_complex > $o,A3: set_complex] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_complex] :
            ( ( X6 @ A5 )
           => ? [X4: complex] :
                ( ( member_complex @ X4 @ A5 )
                & ( ( X6 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2548_infinite__coinduct,axiom,
    ! [X6: set_Pr1261947904930325089at_nat > $o,A3: set_Pr1261947904930325089at_nat] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_Pr1261947904930325089at_nat] :
            ( ( X6 @ A5 )
           => ? [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A5 )
                & ( ( X6 @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) )
                  | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
       => ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2549_infinite__coinduct,axiom,
    ! [X6: set_Extended_enat > $o,A3: set_Extended_enat] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_Extended_enat] :
            ( ( X6 @ A5 )
           => ? [X4: extended_enat] :
                ( ( member_Extended_enat @ X4 @ A5 )
                & ( ( X6 @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) )
                  | ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
       => ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2550_infinite__coinduct,axiom,
    ! [X6: set_real > $o,A3: set_real] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_real] :
            ( ( X6 @ A5 )
           => ? [X4: real] :
                ( ( member_real @ X4 @ A5 )
                & ( ( X6 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2551_infinite__coinduct,axiom,
    ! [X6: set_o > $o,A3: set_o] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_o] :
            ( ( X6 @ A5 )
           => ? [X4: $o] :
                ( ( member_o @ X4 @ A5 )
                & ( ( X6 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2552_infinite__coinduct,axiom,
    ! [X6: set_int > $o,A3: set_int] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_int] :
            ( ( X6 @ A5 )
           => ? [X4: int] :
                ( ( member_int @ X4 @ A5 )
                & ( ( X6 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2553_infinite__coinduct,axiom,
    ! [X6: set_nat > $o,A3: set_nat] :
      ( ( X6 @ A3 )
     => ( ! [A5: set_nat] :
            ( ( X6 @ A5 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ A5 )
                & ( ( X6 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2554_finite__empty__induct,axiom,
    ! [A3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: set_nat,A5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A5 )
             => ( ( member_set_nat @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_2555_finite__empty__induct,axiom,
    ! [A3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: complex,A5: set_complex] :
              ( ( finite3207457112153483333omplex @ A5 )
             => ( ( member_complex @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_2556_finite__empty__induct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A5 )
             => ( ( member8440522571783428010at_nat @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
         => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_2557_finite__empty__induct,axiom,
    ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: extended_enat,A5: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A5 )
             => ( ( member_Extended_enat @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
         => ( P @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% finite_empty_induct
thf(fact_2558_finite__empty__induct,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: real,A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ( member_real @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_2559_finite__empty__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: $o,A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ( member_o @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_2560_finite__empty__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: int,A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ( member_int @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_2561_finite__empty__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A: nat,A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ( member_nat @ A @ A5 )
               => ( ( P @ A5 )
                 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_2562_subset__insert__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
      = ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
         => ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_2563_subset__insert__iff,axiom,
    ! [A3: set_set_nat,X2: set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
      = ( ( ( member_set_nat @ X2 @ A3 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
        & ( ~ ( member_set_nat @ X2 @ A3 )
         => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_2564_subset__insert__iff,axiom,
    ! [A3: set_real,X2: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
      = ( ( ( member_real @ X2 @ A3 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
        & ( ~ ( member_real @ X2 @ A3 )
         => ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_2565_subset__insert__iff,axiom,
    ! [A3: set_o,X2: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
      = ( ( ( member_o @ X2 @ A3 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
        & ( ~ ( member_o @ X2 @ A3 )
         => ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_2566_subset__insert__iff,axiom,
    ! [A3: set_nat,X2: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
      = ( ( ( member_nat @ X2 @ A3 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
        & ( ~ ( member_nat @ X2 @ A3 )
         => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_2567_subset__insert__iff,axiom,
    ! [A3: set_int,X2: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
      = ( ( ( member_int @ X2 @ A3 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 ) )
        & ( ~ ( member_int @ X2 @ A3 )
         => ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_2568_Diff__single__insert,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 )
     => ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_2569_Diff__single__insert,axiom,
    ! [A3: set_real,X2: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 )
     => ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_2570_Diff__single__insert,axiom,
    ! [A3: set_o,X2: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 )
     => ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_2571_Diff__single__insert,axiom,
    ! [A3: set_nat,X2: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 )
     => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_2572_Diff__single__insert,axiom,
    ! [A3: set_int,X2: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 )
     => ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_2573_card__1__singletonE,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = one_one_nat )
     => ~ ! [X5: product_prod_nat_nat] :
            ( A3
           != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).

% card_1_singletonE
thf(fact_2574_card__1__singletonE,axiom,
    ! [A3: set_complex] :
      ( ( ( finite_card_complex @ A3 )
        = one_one_nat )
     => ~ ! [X5: complex] :
            ( A3
           != ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ).

% card_1_singletonE
thf(fact_2575_card__1__singletonE,axiom,
    ! [A3: set_list_nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = one_one_nat )
     => ~ ! [X5: list_nat] :
            ( A3
           != ( insert_list_nat @ X5 @ bot_bot_set_list_nat ) ) ) ).

% card_1_singletonE
thf(fact_2576_card__1__singletonE,axiom,
    ! [A3: set_set_nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = one_one_nat )
     => ~ ! [X5: set_nat] :
            ( A3
           != ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_2577_card__1__singletonE,axiom,
    ! [A3: set_real] :
      ( ( ( finite_card_real @ A3 )
        = one_one_nat )
     => ~ ! [X5: real] :
            ( A3
           != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).

% card_1_singletonE
thf(fact_2578_card__1__singletonE,axiom,
    ! [A3: set_o] :
      ( ( ( finite_card_o @ A3 )
        = one_one_nat )
     => ~ ! [X5: $o] :
            ( A3
           != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_2579_card__1__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = one_one_nat )
     => ~ ! [X5: nat] :
            ( A3
           != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_2580_card__1__singletonE,axiom,
    ! [A3: set_int] :
      ( ( ( finite_card_int @ A3 )
        = one_one_nat )
     => ~ ! [X5: int] :
            ( A3
           != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_2581_Compl__insert,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_2582_Compl__insert,axiom,
    ! [X2: real,A3: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X2 @ A3 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A3 ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_2583_Compl__insert,axiom,
    ! [X2: $o,A3: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X2 @ A3 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_2584_Compl__insert,axiom,
    ! [X2: int,A3: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X2 @ A3 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_2585_Compl__insert,axiom,
    ! [X2: nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X2 @ A3 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_2586_card__Suc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A3
              = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
            & ( ( finite711546835091564841at_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2587_card__Suc__eq,axiom,
    ! [A3: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: complex,B6: set_complex] :
            ( ( A3
              = ( insert_complex @ B4 @ B6 ) )
            & ~ ( member_complex @ B4 @ B6 )
            & ( ( finite_card_complex @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_complex ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2588_card__Suc__eq,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: list_nat,B6: set_list_nat] :
            ( ( A3
              = ( insert_list_nat @ B4 @ B6 ) )
            & ~ ( member_list_nat @ B4 @ B6 )
            & ( ( finite_card_list_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_list_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2589_card__Suc__eq,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: set_nat,B6: set_set_nat] :
            ( ( A3
              = ( insert_set_nat @ B4 @ B6 ) )
            & ~ ( member_set_nat @ B4 @ B6 )
            & ( ( finite_card_set_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2590_card__Suc__eq,axiom,
    ! [A3: set_real,K: nat] :
      ( ( ( finite_card_real @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: real,B6: set_real] :
            ( ( A3
              = ( insert_real @ B4 @ B6 ) )
            & ~ ( member_real @ B4 @ B6 )
            & ( ( finite_card_real @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_real ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2591_card__Suc__eq,axiom,
    ! [A3: set_o,K: nat] :
      ( ( ( finite_card_o @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: $o,B6: set_o] :
            ( ( A3
              = ( insert_o @ B4 @ B6 ) )
            & ~ ( member_o @ B4 @ B6 )
            & ( ( finite_card_o @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_o ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2592_card__Suc__eq,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: nat,B6: set_nat] :
            ( ( A3
              = ( insert_nat @ B4 @ B6 ) )
            & ~ ( member_nat @ B4 @ B6 )
            & ( ( finite_card_nat @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2593_card__Suc__eq,axiom,
    ! [A3: set_int,K: nat] :
      ( ( ( finite_card_int @ A3 )
        = ( suc @ K ) )
      = ( ? [B4: int,B6: set_int] :
            ( ( A3
              = ( insert_int @ B4 @ B6 ) )
            & ~ ( member_int @ B4 @ B6 )
            & ( ( finite_card_int @ B6 )
              = K )
            & ( ( K = zero_zero_nat )
             => ( B6 = bot_bot_set_int ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2594_card__eq__SucD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = ( suc @ K ) )
     => ? [B: product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ( A3
            = ( insert8211810215607154385at_nat @ B @ B8 ) )
          & ~ ( member8440522571783428010at_nat @ B @ B8 )
          & ( ( finite711546835091564841at_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2595_card__eq__SucD,axiom,
    ! [A3: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A3 )
        = ( suc @ K ) )
     => ? [B: complex,B8: set_complex] :
          ( ( A3
            = ( insert_complex @ B @ B8 ) )
          & ~ ( member_complex @ B @ B8 )
          & ( ( finite_card_complex @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
thf(fact_2596_card__eq__SucD,axiom,
    ! [A3: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = ( suc @ K ) )
     => ? [B: list_nat,B8: set_list_nat] :
          ( ( A3
            = ( insert_list_nat @ B @ B8 ) )
          & ~ ( member_list_nat @ B @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2597_card__eq__SucD,axiom,
    ! [A3: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = ( suc @ K ) )
     => ? [B: set_nat,B8: set_set_nat] :
          ( ( A3
            = ( insert_set_nat @ B @ B8 ) )
          & ~ ( member_set_nat @ B @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2598_card__eq__SucD,axiom,
    ! [A3: set_real,K: nat] :
      ( ( ( finite_card_real @ A3 )
        = ( suc @ K ) )
     => ? [B: real,B8: set_real] :
          ( ( A3
            = ( insert_real @ B @ B8 ) )
          & ~ ( member_real @ B @ B8 )
          & ( ( finite_card_real @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
thf(fact_2599_card__eq__SucD,axiom,
    ! [A3: set_o,K: nat] :
      ( ( ( finite_card_o @ A3 )
        = ( suc @ K ) )
     => ? [B: $o,B8: set_o] :
          ( ( A3
            = ( insert_o @ B @ B8 ) )
          & ~ ( member_o @ B @ B8 )
          & ( ( finite_card_o @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_o ) ) ) ) ).

% card_eq_SucD
thf(fact_2600_card__eq__SucD,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A3 )
        = ( suc @ K ) )
     => ? [B: nat,B8: set_nat] :
          ( ( A3
            = ( insert_nat @ B @ B8 ) )
          & ~ ( member_nat @ B @ B8 )
          & ( ( finite_card_nat @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2601_card__eq__SucD,axiom,
    ! [A3: set_int,K: nat] :
      ( ( ( finite_card_int @ A3 )
        = ( suc @ K ) )
     => ? [B: int,B8: set_int] :
          ( ( A3
            = ( insert_int @ B @ B8 ) )
          & ~ ( member_int @ B @ B8 )
          & ( ( finite_card_int @ B8 )
            = K )
          & ( ( K = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
thf(fact_2602_card__1__singleton__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: product_prod_nat_nat] :
            ( A3
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2603_card__1__singleton__iff,axiom,
    ! [A3: set_complex] :
      ( ( ( finite_card_complex @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: complex] :
            ( A3
            = ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2604_card__1__singleton__iff,axiom,
    ! [A3: set_list_nat] :
      ( ( ( finite_card_list_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: list_nat] :
            ( A3
            = ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2605_card__1__singleton__iff,axiom,
    ! [A3: set_set_nat] :
      ( ( ( finite_card_set_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: set_nat] :
            ( A3
            = ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2606_card__1__singleton__iff,axiom,
    ! [A3: set_real] :
      ( ( ( finite_card_real @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: real] :
            ( A3
            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2607_card__1__singleton__iff,axiom,
    ! [A3: set_o] :
      ( ( ( finite_card_o @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: $o] :
            ( A3
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2608_card__1__singleton__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: nat] :
            ( A3
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2609_card__1__singleton__iff,axiom,
    ! [A3: set_int] :
      ( ( ( finite_card_int @ A3 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X: int] :
            ( A3
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2610_remove__induct,axiom,
    ! [P: set_set_nat > $o,B2: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A5 )
             => ( ( A5 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
                 => ( ! [X4: set_nat] :
                        ( ( member_set_nat @ X4 @ A5 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2611_remove__induct,axiom,
    ! [P: set_complex > $o,B2: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_complex] :
              ( ( finite3207457112153483333omplex @ A5 )
             => ( ( A5 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
                 => ( ! [X4: complex] :
                        ( ( member_complex @ X4 @ A5 )
                       => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2612_remove__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
      ( ( P @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A5 )
             => ( ( A5 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
                 => ( ! [X4: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X4 @ A5 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2613_remove__induct,axiom,
    ! [P: set_Extended_enat > $o,B2: set_Extended_enat] :
      ( ( P @ bot_bo7653980558646680370d_enat )
     => ( ( ~ ( finite4001608067531595151d_enat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A5 )
             => ( ( A5 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A5 @ B2 )
                 => ( ! [X4: extended_enat] :
                        ( ( member_Extended_enat @ X4 @ A5 )
                       => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2614_remove__induct,axiom,
    ! [P: set_real > $o,B2: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ( A5 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A5 @ B2 )
                 => ( ! [X4: real] :
                        ( ( member_real @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2615_remove__induct,axiom,
    ! [P: set_o > $o,B2: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ( A5 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A5 @ B2 )
                 => ( ! [X4: $o] :
                        ( ( member_o @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2616_remove__induct,axiom,
    ! [P: set_nat > $o,B2: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ( A5 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A5 @ B2 )
                 => ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2617_remove__induct,axiom,
    ! [P: set_int > $o,B2: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B2 )
         => ( P @ B2 ) )
       => ( ! [A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ( A5 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A5 @ B2 )
                 => ( ! [X4: int] :
                        ( ( member_int @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_2618_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A5 )
             => ( ( A5 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
                 => ( ! [X4: set_nat] :
                        ( ( member_set_nat @ X4 @ A5 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2619_finite__remove__induct,axiom,
    ! [B2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A5: set_complex] :
              ( ( finite3207457112153483333omplex @ A5 )
             => ( ( A5 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
                 => ( ! [X4: complex] :
                        ( ( member_complex @ X4 @ A5 )
                       => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2620_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [A5: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A5 )
             => ( ( A5 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
                 => ( ! [X4: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X4 @ A5 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2621_finite__remove__induct,axiom,
    ! [B2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [A5: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A5 )
             => ( ( A5 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A5 @ B2 )
                 => ( ! [X4: extended_enat] :
                        ( ( member_Extended_enat @ X4 @ A5 )
                       => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2622_finite__remove__induct,axiom,
    ! [B2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A5: set_real] :
              ( ( finite_finite_real @ A5 )
             => ( ( A5 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A5 @ B2 )
                 => ( ! [X4: real] :
                        ( ( member_real @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2623_finite__remove__induct,axiom,
    ! [B2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A5: set_o] :
              ( ( finite_finite_o @ A5 )
             => ( ( A5 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A5 @ B2 )
                 => ( ! [X4: $o] :
                        ( ( member_o @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2624_finite__remove__induct,axiom,
    ! [B2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A5: set_nat] :
              ( ( finite_finite_nat @ A5 )
             => ( ( A5 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A5 @ B2 )
                 => ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2625_finite__remove__induct,axiom,
    ! [B2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A5: set_int] :
              ( ( finite_finite_int @ A5 )
             => ( ( A5 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A5 @ B2 )
                 => ( ! [X4: int] :
                        ( ( member_int @ X4 @ A5 )
                       => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
                   => ( P @ A5 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_2626_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_real] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A3 ) )
      = ( ? [A4: real,B6: set_real] :
            ( ( A3
              = ( insert_real @ A4 @ B6 ) )
            & ~ ( member_real @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_real @ B6 ) )
            & ( finite_finite_real @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2627_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_o @ A3 ) )
      = ( ? [A4: $o,B6: set_o] :
            ( ( A3
              = ( insert_o @ A4 @ B6 ) )
            & ~ ( member_o @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_o @ B6 ) )
            & ( finite_finite_o @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2628_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A3 ) )
      = ( ? [A4: list_nat,B6: set_list_nat] :
            ( ( A3
              = ( insert_list_nat @ A4 @ B6 ) )
            & ~ ( member_list_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B6 ) )
            & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2629_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_nat @ A3 ) )
      = ( ? [A4: set_nat,B6: set_set_nat] :
            ( ( A3
              = ( insert_set_nat @ A4 @ B6 ) )
            & ~ ( member_set_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ B6 ) )
            & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2630_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A3 ) )
      = ( ? [A4: nat,B6: set_nat] :
            ( ( A3
              = ( insert_nat @ A4 @ B6 ) )
            & ~ ( member_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
            & ( finite_finite_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2631_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A3 ) )
      = ( ? [A4: int,B6: set_int] :
            ( ( A3
              = ( insert_int @ A4 @ B6 ) )
            & ~ ( member_int @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_int @ B6 ) )
            & ( finite_finite_int @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2632_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_complex] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_complex @ A3 ) )
      = ( ? [A4: complex,B6: set_complex] :
            ( ( A3
              = ( insert_complex @ A4 @ B6 ) )
            & ~ ( member_complex @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite_card_complex @ B6 ) )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2633_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ A3 ) )
      = ( ? [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
            ( ( A3
              = ( insert8211810215607154385at_nat @ A4 @ B6 ) )
            & ~ ( member8440522571783428010at_nat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ B6 ) )
            & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2634_card__le__Suc__iff,axiom,
    ! [N: nat,A3: set_Extended_enat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ A3 ) )
      = ( ? [A4: extended_enat,B6: set_Extended_enat] :
            ( ( A3
              = ( insert_Extended_enat @ A4 @ B6 ) )
            & ~ ( member_Extended_enat @ A4 @ B6 )
            & ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ B6 ) )
            & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2635_card__Diff1__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ).

% card_Diff1_le
thf(fact_2636_card__Diff1__le,axiom,
    ! [A3: set_complex,X2: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ).

% card_Diff1_le
thf(fact_2637_card__Diff1__le,axiom,
    ! [A3: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ).

% card_Diff1_le
thf(fact_2638_card__Diff1__le,axiom,
    ! [A3: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ).

% card_Diff1_le
thf(fact_2639_card__Diff1__le,axiom,
    ! [A3: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ).

% card_Diff1_le
thf(fact_2640_card__Diff1__le,axiom,
    ! [A3: set_o,X2: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ).

% card_Diff1_le
thf(fact_2641_card__Diff1__le,axiom,
    ! [A3: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ).

% card_Diff1_le
thf(fact_2642_card__Diff1__le,axiom,
    ! [A3: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ).

% card_Diff1_le
thf(fact_2643_finite__induct__select,axiom,
    ! [S: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T4: set_complex] :
              ( ( ord_less_set_complex @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: complex] :
                    ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ S @ T4 ) )
                    & ( P @ ( insert_complex @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2644_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ ( minus_1356011639430497352at_nat @ S @ T4 ) )
                    & ( P @ ( insert8211810215607154385at_nat @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2645_finite__induct__select,axiom,
    ! [S: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [T4: set_Extended_enat] :
              ( ( ord_le2529575680413868914d_enat @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: extended_enat] :
                    ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ S @ T4 ) )
                    & ( P @ ( insert_Extended_enat @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2646_finite__induct__select,axiom,
    ! [S: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T4: set_real] :
              ( ( ord_less_set_real @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: real] :
                    ( ( member_real @ X4 @ ( minus_minus_set_real @ S @ T4 ) )
                    & ( P @ ( insert_real @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2647_finite__induct__select,axiom,
    ! [S: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T4: set_o] :
              ( ( ord_less_set_o @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: $o] :
                    ( ( member_o @ X4 @ ( minus_minus_set_o @ S @ T4 ) )
                    & ( P @ ( insert_o @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2648_finite__induct__select,axiom,
    ! [S: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T4: set_int] :
              ( ( ord_less_set_int @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: int] :
                    ( ( member_int @ X4 @ ( minus_minus_set_int @ S @ T4 ) )
                    & ( P @ ( insert_int @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2649_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S )
             => ( ( P @ T4 )
               => ? [X4: nat] :
                    ( ( member_nat @ X4 @ ( minus_minus_set_nat @ S @ T4 ) )
                    & ( P @ ( insert_nat @ X4 @ T4 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2650_psubset__insert__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
      = ( ( ( member8440522571783428010at_nat @ X2 @ B2 )
         => ( ord_le7866589430770878221at_nat @ A3 @ B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
         => ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
            & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
             => ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2651_psubset__insert__iff,axiom,
    ! [A3: set_set_nat,X2: set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
      = ( ( ( member_set_nat @ X2 @ B2 )
         => ( ord_less_set_set_nat @ A3 @ B2 ) )
        & ( ~ ( member_set_nat @ X2 @ B2 )
         => ( ( ( member_set_nat @ X2 @ A3 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
            & ( ~ ( member_set_nat @ X2 @ A3 )
             => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2652_psubset__insert__iff,axiom,
    ! [A3: set_real,X2: real,B2: set_real] :
      ( ( ord_less_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
      = ( ( ( member_real @ X2 @ B2 )
         => ( ord_less_set_real @ A3 @ B2 ) )
        & ( ~ ( member_real @ X2 @ B2 )
         => ( ( ( member_real @ X2 @ A3 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
            & ( ~ ( member_real @ X2 @ A3 )
             => ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2653_psubset__insert__iff,axiom,
    ! [A3: set_o,X2: $o,B2: set_o] :
      ( ( ord_less_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
      = ( ( ( member_o @ X2 @ B2 )
         => ( ord_less_set_o @ A3 @ B2 ) )
        & ( ~ ( member_o @ X2 @ B2 )
         => ( ( ( member_o @ X2 @ A3 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
            & ( ~ ( member_o @ X2 @ A3 )
             => ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2654_psubset__insert__iff,axiom,
    ! [A3: set_nat,X2: nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
      = ( ( ( member_nat @ X2 @ B2 )
         => ( ord_less_set_nat @ A3 @ B2 ) )
        & ( ~ ( member_nat @ X2 @ B2 )
         => ( ( ( member_nat @ X2 @ A3 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
            & ( ~ ( member_nat @ X2 @ A3 )
             => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2655_psubset__insert__iff,axiom,
    ! [A3: set_int,X2: int,B2: set_int] :
      ( ( ord_less_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
      = ( ( ( member_int @ X2 @ B2 )
         => ( ord_less_set_int @ A3 @ B2 ) )
        & ( ~ ( member_int @ X2 @ B2 )
         => ( ( ( member_int @ X2 @ A3 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 ) )
            & ( ~ ( member_int @ X2 @ A3 )
             => ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2656_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X5: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2657_card_Oremove,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( member_list_nat @ X2 @ A3 )
       => ( ( finite_card_list_nat @ A3 )
          = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2658_card_Oremove,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X2 @ A3 )
       => ( ( finite_card_set_nat @ A3 )
          = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2659_card_Oremove,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( finite_card_complex @ A3 )
          = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2660_card_Oremove,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ( finite711546835091564841at_nat @ A3 )
          = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2661_card_Oremove,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( finite121521170596916366d_enat @ A3 )
          = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2662_card_Oremove,axiom,
    ! [A3: set_real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ( finite_card_real @ A3 )
          = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2663_card_Oremove,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X2 @ A3 )
       => ( ( finite_card_o @ A3 )
          = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2664_card_Oremove,axiom,
    ! [A3: set_int,X2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X2 @ A3 )
       => ( ( finite_card_int @ A3 )
          = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2665_card_Oremove,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X2 @ A3 )
       => ( ( finite_card_nat @ A3 )
          = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2666_card_Oinsert__remove,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
        = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2667_card_Oinsert__remove,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
        = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2668_card_Oinsert__remove,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
        = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2669_card_Oinsert__remove,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
        = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2670_card_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
        = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2671_card_Oinsert__remove,axiom,
    ! [A3: set_real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
        = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2672_card_Oinsert__remove,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
        = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2673_card_Oinsert__remove,axiom,
    ! [A3: set_int,X2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
        = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2674_card_Oinsert__remove,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
        = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2675_card__Suc__Diff1,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( member_list_nat @ X2 @ A3 )
       => ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) )
          = ( finite_card_list_nat @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2676_card__Suc__Diff1,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X2 @ A3 )
       => ( ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
          = ( finite_card_set_nat @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2677_card__Suc__Diff1,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
          = ( finite_card_complex @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2678_card__Suc__Diff1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
          = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2679_card__Suc__Diff1,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
          = ( finite121521170596916366d_enat @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2680_card__Suc__Diff1,axiom,
    ! [A3: set_real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
          = ( finite_card_real @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2681_card__Suc__Diff1,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X2 @ A3 )
       => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) )
          = ( finite_card_o @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2682_card__Suc__Diff1,axiom,
    ! [A3: set_int,X2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X2 @ A3 )
       => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
          = ( finite_card_int @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2683_card__Suc__Diff1,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X2 @ A3 )
       => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2684_card__Diff1__less,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( member_list_nat @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2685_card__Diff1__less,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2686_card__Diff1__less,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2687_card__Diff1__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2688_card__Diff1__less,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2689_card__Diff1__less,axiom,
    ! [A3: set_real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2690_card__Diff1__less,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2691_card__Diff1__less,axiom,
    ! [A3: set_int,X2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2692_card__Diff1__less,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X2 @ A3 )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_2693_card__Diff2__less,axiom,
    ! [A3: set_list_nat,X2: list_nat,Y3: list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( member_list_nat @ X2 @ A3 )
       => ( ( member_list_nat @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ ( insert_list_nat @ Y3 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2694_card__Diff2__less,axiom,
    ! [A3: set_set_nat,X2: set_nat,Y3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X2 @ A3 )
       => ( ( member_set_nat @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2695_card__Diff2__less,axiom,
    ! [A3: set_complex,X2: complex,Y3: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( member_complex @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) @ ( insert_complex @ Y3 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2696_card__Diff2__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2697_card__Diff2__less,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,Y3: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( member_Extended_enat @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) @ ( insert_Extended_enat @ Y3 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2698_card__Diff2__less,axiom,
    ! [A3: set_real,X2: real,Y3: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ( member_real @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ ( insert_real @ Y3 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2699_card__Diff2__less,axiom,
    ! [A3: set_o,X2: $o,Y3: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X2 @ A3 )
       => ( ( member_o @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ ( insert_o @ Y3 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2700_card__Diff2__less,axiom,
    ! [A3: set_int,X2: int,Y3: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( member_int @ X2 @ A3 )
       => ( ( member_int @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ ( insert_int @ Y3 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2701_card__Diff2__less,axiom,
    ! [A3: set_nat,X2: nat,Y3: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ X2 @ A3 )
       => ( ( member_nat @ Y3 @ A3 )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_2702_card__Diff1__less__iff,axiom,
    ! [A3: set_list_nat,X2: list_nat] :
      ( ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) )
      = ( ( finite8100373058378681591st_nat @ A3 )
        & ( member_list_nat @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2703_card__Diff1__less__iff,axiom,
    ! [A3: set_set_nat,X2: set_nat] :
      ( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) )
      = ( ( finite1152437895449049373et_nat @ A3 )
        & ( member_set_nat @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2704_card__Diff1__less__iff,axiom,
    ! [A3: set_complex,X2: complex] :
      ( ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) )
      = ( ( finite3207457112153483333omplex @ A3 )
        & ( member_complex @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2705_card__Diff1__less__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) )
      = ( ( finite6177210948735845034at_nat @ A3 )
        & ( member8440522571783428010at_nat @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2706_card__Diff1__less__iff,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat] :
      ( ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) )
      = ( ( finite4001608067531595151d_enat @ A3 )
        & ( member_Extended_enat @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2707_card__Diff1__less__iff,axiom,
    ! [A3: set_real,X2: real] :
      ( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) )
      = ( ( finite_finite_real @ A3 )
        & ( member_real @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2708_card__Diff1__less__iff,axiom,
    ! [A3: set_o,X2: $o] :
      ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) )
      = ( ( finite_finite_o @ A3 )
        & ( member_o @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2709_card__Diff1__less__iff,axiom,
    ! [A3: set_int,X2: int] :
      ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) )
      = ( ( finite_finite_int @ A3 )
        & ( member_int @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2710_card__Diff1__less__iff,axiom,
    ! [A3: set_nat,X2: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) )
      = ( ( finite_finite_nat @ A3 )
        & ( member_nat @ X2 @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_2711_card__Diff__singleton,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ A3 )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2712_card__Diff__singleton,axiom,
    ! [X2: complex,A3: set_complex] :
      ( ( member_complex @ X2 @ A3 )
     => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
        = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2713_card__Diff__singleton,axiom,
    ! [X2: list_nat,A3: set_list_nat] :
      ( ( member_list_nat @ X2 @ A3 )
     => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2714_card__Diff__singleton,axiom,
    ! [X2: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X2 @ A3 )
     => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2715_card__Diff__singleton,axiom,
    ! [X2: real,A3: set_real] :
      ( ( member_real @ X2 @ A3 )
     => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
        = ( minus_minus_nat @ ( finite_card_real @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2716_card__Diff__singleton,axiom,
    ! [X2: $o,A3: set_o] :
      ( ( member_o @ X2 @ A3 )
     => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
        = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2717_card__Diff__singleton,axiom,
    ! [X2: int,A3: set_int] :
      ( ( member_int @ X2 @ A3 )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
        = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2718_card__Diff__singleton,axiom,
    ! [X2: nat,A3: set_nat] :
      ( ( member_nat @ X2 @ A3 )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_2719_card__Diff__singleton__if,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
          = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2720_card__Diff__singleton__if,axiom,
    ! [X2: complex,A3: set_complex] :
      ( ( ( member_complex @ X2 @ A3 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_complex @ X2 @ A3 )
       => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
          = ( finite_card_complex @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2721_card__Diff__singleton__if,axiom,
    ! [X2: list_nat,A3: set_list_nat] :
      ( ( ( member_list_nat @ X2 @ A3 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_list_nat @ X2 @ A3 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
          = ( finite_card_list_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2722_card__Diff__singleton__if,axiom,
    ! [X2: set_nat,A3: set_set_nat] :
      ( ( ( member_set_nat @ X2 @ A3 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat @ X2 @ A3 )
       => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
          = ( finite_card_set_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2723_card__Diff__singleton__if,axiom,
    ! [X2: real,A3: set_real] :
      ( ( ( member_real @ X2 @ A3 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( finite_card_real @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_real @ X2 @ A3 )
       => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
          = ( finite_card_real @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2724_card__Diff__singleton__if,axiom,
    ! [X2: $o,A3: set_o] :
      ( ( ( member_o @ X2 @ A3 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X2 @ A3 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
          = ( finite_card_o @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2725_card__Diff__singleton__if,axiom,
    ! [X2: int,A3: set_int] :
      ( ( ( member_int @ X2 @ A3 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X2 @ A3 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
          = ( finite_card_int @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2726_card__Diff__singleton__if,axiom,
    ! [X2: nat,A3: set_nat] :
      ( ( ( member_nat @ X2 @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X2 @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_2727_enumerate__Suc_H,axiom,
    ! [S: set_nat,N: nat] :
      ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
      = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).

% enumerate_Suc'
thf(fact_2728_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2729_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_real @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_real @ ( insert_real @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2730_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_o,X2: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2731_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_complex,X2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2732_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_list_nat,X2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2733_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_set_nat,X2: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2734_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2735_card__insert__le__m1,axiom,
    ! [N: nat,Y3: set_int,X2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X2 @ Y3 ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2736_vebt__pred_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A: $o,Uw2: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A: $o,B: $o,Va2: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_2737_vebt__succ_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_2738_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X5 ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2739_vebt__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X5: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_2740_vebt__delete_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) )
     => ( ! [A: $o,B: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A: $o,B: $o,N3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) ) )
         => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X5 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X5 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_2741_vebt__insert_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X5: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
     => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X5 ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X5 ) )
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_2742_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_2743_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_2744_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_2745_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_2746_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).

% List.finite_set
thf(fact_2747_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs ) ) ).

% List.finite_set
thf(fact_2748_Bolzano,axiom,
    ! [A2: real,B3: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ! [A: real,B: real,C3: real] :
            ( ( P @ A @ B )
           => ( ( P @ B @ C3 )
             => ( ( ord_less_eq_real @ A @ B )
               => ( ( ord_less_eq_real @ B @ C3 )
                 => ( P @ A @ C3 ) ) ) ) )
       => ( ! [X5: real] :
              ( ( ord_less_eq_real @ A2 @ X5 )
             => ( ( ord_less_eq_real @ X5 @ B3 )
               => ? [D3: real] :
                    ( ( ord_less_real @ zero_zero_real @ D3 )
                    & ! [A: real,B: real] :
                        ( ( ( ord_less_eq_real @ A @ X5 )
                          & ( ord_less_eq_real @ X5 @ B )
                          & ( ord_less_real @ ( minus_minus_real @ B @ A ) @ D3 ) )
                       => ( P @ A @ B ) ) ) ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% Bolzano
thf(fact_2749_card__length,axiom,
    ! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).

% card_length
thf(fact_2750_card__length,axiom,
    ! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).

% card_length
thf(fact_2751_card__length,axiom,
    ! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).

% card_length
thf(fact_2752_card__length,axiom,
    ! [Xs: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).

% card_length
thf(fact_2753_card__length,axiom,
    ! [Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% card_length
thf(fact_2754_card__length,axiom,
    ! [Xs: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).

% card_length
thf(fact_2755_card__length,axiom,
    ! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% card_length
thf(fact_2756_length__pos__if__in__set,axiom,
    ! [X2: real,Xs: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2757_length__pos__if__in__set,axiom,
    ! [X2: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2758_length__pos__if__in__set,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2759_length__pos__if__in__set,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2760_length__pos__if__in__set,axiom,
    ! [X2: $o,Xs: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2761_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2762_the__elem__eq,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
      = X2 ) ).

% the_elem_eq
thf(fact_2763_the__elem__eq,axiom,
    ! [X2: real] :
      ( ( the_elem_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
      = X2 ) ).

% the_elem_eq
thf(fact_2764_the__elem__eq,axiom,
    ! [X2: $o] :
      ( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
      = X2 ) ).

% the_elem_eq
thf(fact_2765_the__elem__eq,axiom,
    ! [X2: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
      = X2 ) ).

% the_elem_eq
thf(fact_2766_the__elem__eq,axiom,
    ! [X2: int] :
      ( ( the_elem_int @ ( insert_int @ X2 @ bot_bot_set_int ) )
      = X2 ) ).

% the_elem_eq
thf(fact_2767_case4_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList2 ) )
     => ( ( vEBT_invar_vebt @ X4 @ na )
        & ! [Xa: vEBT_VEBT] :
            ( ( vEBT_invar_vebt @ Xa @ na )
           => ( ( ( vEBT_VEBT_set_vebt @ X4 )
                = ( vEBT_VEBT_set_vebt @ Xa ) )
             => ( Xa = X4 ) ) ) ) ) ).

% case4(1)
thf(fact_2768_vebt__maxt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( ( B
                   => ( Y3
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B
                   => ( ( A
                       => ( Y3
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_2769_vebt__mint_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( ( A
                   => ( Y3
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A
                   => ( ( B
                       => ( Y3
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_2770_is__singletonI,axiom,
    ! [X2: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_2771_is__singletonI,axiom,
    ! [X2: real] : ( is_singleton_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_2772_is__singletonI,axiom,
    ! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_2773_is__singletonI,axiom,
    ! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_2774_is__singletonI,axiom,
    ! [X2: int] : ( is_singleton_int @ ( insert_int @ X2 @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_2775_case4_I5_J,axiom,
    ( m
    = ( suc @ na ) ) ).

% case4(5)
thf(fact_2776_is__singleton__the__elem,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2777_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A6: set_real] :
          ( A6
          = ( insert_real @ ( the_elem_real @ A6 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2778_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
          ( A6
          = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2779_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( A6
          = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2780_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
          ( A6
          = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2781_is__singletonI_H,axiom,
    ! [A3: set_set_nat] :
      ( ( A3 != bot_bot_set_set_nat )
     => ( ! [X5: set_nat,Y4: set_nat] :
            ( ( member_set_nat @ X5 @ A3 )
           => ( ( member_set_nat @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_set_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2782_is__singletonI_H,axiom,
    ! [A3: set_real] :
      ( ( A3 != bot_bot_set_real )
     => ( ! [X5: real,Y4: real] :
            ( ( member_real @ X5 @ A3 )
           => ( ( member_real @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_real @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2783_is__singletonI_H,axiom,
    ! [A3: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X5: $o,Y4: $o] :
            ( ( member_o @ X5 @ A3 )
           => ( ( member_o @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_o @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2784_is__singletonI_H,axiom,
    ! [A3: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X5: nat,Y4: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ( member_nat @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2785_is__singletonI_H,axiom,
    ! [A3: set_int] :
      ( ( A3 != bot_bot_set_int )
     => ( ! [X5: int,Y4: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ( member_int @ Y4 @ A3 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_int @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2786_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_2787_length__induct,axiom,
    ! [P: list_o > $o,Xs: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_2788_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_2789_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M5 )
     => ? [N3: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X4 @ M5 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2790_finite__maxlen,axiom,
    ! [M5: set_list_o] :
      ( ( finite_finite_list_o @ M5 )
     => ? [N3: nat] :
        ! [X4: list_o] :
          ( ( member_list_o @ X4 @ M5 )
         => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2791_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M5 )
     => ? [N3: nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ M5 )
         => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2792_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [X: product_prod_nat_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_2793_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A6: set_real] :
        ? [X: real] :
          ( A6
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_2794_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A6: set_o] :
        ? [X: $o] :
          ( A6
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_2795_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
        ? [X: nat] :
          ( A6
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_2796_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
        ? [X: int] :
          ( A6
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_2797_is__singletonE,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A3 )
     => ~ ! [X5: product_prod_nat_nat] :
            ( A3
           != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_2798_is__singletonE,axiom,
    ! [A3: set_real] :
      ( ( is_singleton_real @ A3 )
     => ~ ! [X5: real] :
            ( A3
           != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_2799_is__singletonE,axiom,
    ! [A3: set_o] :
      ( ( is_singleton_o @ A3 )
     => ~ ! [X5: $o] :
            ( A3
           != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_2800_is__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( is_singleton_nat @ A3 )
     => ~ ! [X5: nat] :
            ( A3
           != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_2801_is__singletonE,axiom,
    ! [A3: set_int] :
      ( ( is_singleton_int @ A3 )
     => ~ ! [X5: int] :
            ( A3
           != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_2802_is__singleton__altdef,axiom,
    ( is_singleton_complex
    = ( ^ [A6: set_complex] :
          ( ( finite_card_complex @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_2803_is__singleton__altdef,axiom,
    ( is_sin2641923865335537900st_nat
    = ( ^ [A6: set_list_nat] :
          ( ( finite_card_list_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_2804_is__singleton__altdef,axiom,
    ( is_singleton_set_nat
    = ( ^ [A6: set_set_nat] :
          ( ( finite_card_set_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_2805_is__singleton__altdef,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
          ( ( finite_card_nat @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_2806_is__singleton__altdef,axiom,
    ( is_singleton_int
    = ( ^ [A6: set_int] :
          ( ( finite_card_int @ A6 )
          = one_one_nat ) ) ) ).

% is_singleton_altdef
thf(fact_2807_finite__list,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_2808_finite__list,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_2809_finite__list,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_2810_finite__list,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_2811_finite__list,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ? [Xs3: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_2812_finite__list,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ? [Xs3: list_Extended_enat] :
          ( ( set_Extended_enat2 @ Xs3 )
          = A3 ) ) ).

% finite_list
thf(fact_2813_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs ) )
           => ( member_real @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2814_subset__code_I1_J,axiom,
    ! [Xs: list_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
      = ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs ) )
           => ( member_o @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2815_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2816_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2817_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2818_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs ) )
           => ( member_int @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2819_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X4: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList2 ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ).

% case4(8)
thf(fact_2820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y3: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y3
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y3
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y3
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y3
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y3
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_2821_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_2822_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_2823_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ! [Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_2824_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_2825_set__removeAll,axiom,
    ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( remove3673390508374433037at_nat @ X2 @ Xs ) )
      = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% set_removeAll
thf(fact_2826_set__removeAll,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) )
      = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_removeAll
thf(fact_2827_set__removeAll,axiom,
    ! [X2: real,Xs: list_real] :
      ( ( set_real2 @ ( removeAll_real @ X2 @ Xs ) )
      = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).

% set_removeAll
thf(fact_2828_set__removeAll,axiom,
    ! [X2: $o,Xs: list_o] :
      ( ( set_o2 @ ( removeAll_o @ X2 @ Xs ) )
      = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).

% set_removeAll
thf(fact_2829_set__removeAll,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( set_int2 @ ( removeAll_int @ X2 @ Xs ) )
      = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).

% set_removeAll
thf(fact_2830_set__removeAll,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) )
      = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% set_removeAll
thf(fact_2831_inthall,axiom,
    ! [Xs: list_real,P: real > $o,N: nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_real2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_2832_inthall,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_2833_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N: nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_2834_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_2835_inthall,axiom,
    ! [Xs: list_o,P: $o > $o,N: nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_2836_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_2837_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_2838_case4_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% case4(9)
thf(fact_2839_case4_I3_J,axiom,
    vEBT_invar_vebt @ summary2 @ m ).

% case4(3)
thf(fact_2840_abs__idempotent,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_idempotent
thf(fact_2841_abs__idempotent,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_idempotent
thf(fact_2842_abs__idempotent,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_idempotent
thf(fact_2843_abs__idempotent,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_idempotent
thf(fact_2844_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_2845_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_2846_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_2847_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_2848_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_2849_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_2850_abs__0__eq,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_2851_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_2852_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_2853_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_2854_abs__eq__0,axiom,
    ! [A2: real] :
      ( ( ( abs_abs_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_2855_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_2856_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_2857_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_2858_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_2859_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_2860_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_2861_abs__minus__cancel,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_minus_cancel
thf(fact_2862_abs__minus__cancel,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_minus_cancel
thf(fact_2863_abs__minus__cancel,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_minus_cancel
thf(fact_2864_abs__minus__cancel,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_minus_cancel
thf(fact_2865_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_2866_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_2867_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_2868_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_2869_abs__le__zero__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_2870_abs__le__zero__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_2871_abs__le__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_2872_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_2873_abs__le__self__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ A2 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% abs_le_self_iff
thf(fact_2874_abs__le__self__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ A2 )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% abs_le_self_iff
thf(fact_2875_abs__le__self__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ A2 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% abs_le_self_iff
thf(fact_2876_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% abs_le_self_iff
thf(fact_2877_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_2878_abs__of__nonneg,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( abs_abs_real @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_2879_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_2880_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_2881_zero__less__abs__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) )
      = ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_2882_zero__less__abs__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A2 ) )
      = ( A2 != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_2883_zero__less__abs__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) )
      = ( A2 != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_2884_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
      = ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_2885_zero__le__divide__abs__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B3 ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
        | ( B3 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_2886_zero__le__divide__abs__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B3 ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
        | ( B3 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_2887_divide__le__0__abs__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B3 ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A2 @ zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_2888_divide__le__0__abs__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B3 ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_2889_abs__of__nonpos,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( abs_abs_real @ A2 )
        = ( uminus_uminus_real @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_2890_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_2891_abs__of__nonpos,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A2 )
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_2892_abs__of__nonpos,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( abs_abs_int @ A2 )
        = ( uminus_uminus_int @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_2893_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_2894_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_2895_case4_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% case4(6)
thf(fact_2896_abs__le__D1,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
     => ( ord_less_eq_real @ A2 @ B3 ) ) ).

% abs_le_D1
thf(fact_2897_abs__le__D1,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
     => ( ord_le3102999989581377725nteger @ A2 @ B3 ) ) ).

% abs_le_D1
thf(fact_2898_abs__le__D1,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
     => ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% abs_le_D1
thf(fact_2899_abs__le__D1,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
     => ( ord_less_eq_int @ A2 @ B3 ) ) ).

% abs_le_D1
thf(fact_2900_abs__ge__self,axiom,
    ! [A2: real] : ( ord_less_eq_real @ A2 @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_self
thf(fact_2901_abs__ge__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_self
thf(fact_2902_abs__ge__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_self
thf(fact_2903_abs__ge__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_self
thf(fact_2904_abs__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_2905_abs__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( abs_abs_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_2906_abs__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_2907_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_2908_abs__minus__commute,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_2909_abs__minus__commute,axiom,
    ! [A2: real,B3: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) )
      = ( abs_abs_real @ ( minus_minus_real @ B3 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_2910_abs__minus__commute,axiom,
    ! [A2: rat,B3: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_2911_abs__minus__commute,axiom,
    ! [A2: int,B3: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) )
      = ( abs_abs_int @ ( minus_minus_int @ B3 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_2912_infinite__int__iff__unbounded__le,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ! [M2: int] :
          ? [N2: int] :
            ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_2913_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ! [M2: int] :
          ? [N2: int] :
            ( ( ord_less_int @ M2 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_2914_abs__ge__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_zero
thf(fact_2915_abs__ge__zero,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_zero
thf(fact_2916_abs__ge__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_zero
thf(fact_2917_abs__ge__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_zero
thf(fact_2918_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_2919_abs__of__pos,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( abs_abs_real @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_2920_abs__of__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_2921_abs__of__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_2922_abs__not__less__zero,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_2923_abs__not__less__zero,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A2 ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_2924_abs__not__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_2925_abs__not__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_2926_abs__triangle__ineq2__sym,axiom,
    ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_2927_abs__triangle__ineq2__sym,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_2928_abs__triangle__ineq2__sym,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_2929_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_2930_abs__triangle__ineq3,axiom,
    ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_2931_abs__triangle__ineq3,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_2932_abs__triangle__ineq3,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_2933_abs__triangle__ineq3,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_2934_abs__triangle__ineq2,axiom,
    ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_2935_abs__triangle__ineq2,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_2936_abs__triangle__ineq2,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_2937_abs__triangle__ineq2,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_2938_nonzero__abs__divide,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A2 @ B3 ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_2939_nonzero__abs__divide,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A2 @ B3 ) )
        = ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_2940_abs__ge__minus__self,axiom,
    ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_2941_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_2942_abs__ge__minus__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_2943_abs__ge__minus__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_2944_abs__le__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
      = ( ( ord_less_eq_real @ A2 @ B3 )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_2945_abs__le__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
      = ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_2946_abs__le__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
      = ( ( ord_less_eq_rat @ A2 @ B3 )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_2947_abs__le__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
      = ( ( ord_less_eq_int @ A2 @ B3 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_2948_abs__le__D2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_2949_abs__le__D2,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_2950_abs__le__D2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_2951_abs__le__D2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_2952_abs__leI,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 )
       => ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_2953_abs__leI,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_2954_abs__leI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_2955_abs__leI,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_2956_abs__less__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ B3 )
      = ( ( ord_less_int @ A2 @ B3 )
        & ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_2957_abs__less__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ B3 )
      = ( ( ord_less_real @ A2 @ B3 )
        & ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_2958_abs__less__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ B3 )
      = ( ( ord_less_rat @ A2 @ B3 )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_2959_abs__less__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
      = ( ( ord_le6747313008572928689nteger @ A2 @ B3 )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_2960_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I2 )
              = ( nth_int @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_2961_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I2 )
              = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_2962_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I2 )
              = ( nth_o @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_2963_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys2 @ I2 ) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_2964_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: int] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2965_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: vEBT_VEBT] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2966_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: $o] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_o] :
            ( ( ( size_size_list_o @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2967_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: nat] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2968_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z3: list_int] : Y6 = Z3 )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I4 )
                = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2969_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : Y6 = Z3 )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
                = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2970_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z3: list_o] : Y6 = Z3 )
    = ( ^ [Xs2: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
             => ( ( nth_o @ Xs2 @ I4 )
                = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2971_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z3: list_nat] : Y6 = Z3 )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2972_dense__eq0__I,axiom,
    ! [X2: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E ) )
     => ( X2 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_2973_dense__eq0__I,axiom,
    ! [X2: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E ) )
     => ( X2 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_2974_eq__abs__iff_H,axiom,
    ! [A2: real,B3: real] :
      ( ( A2
        = ( abs_abs_real @ B3 ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
        & ( ( B3 = A2 )
          | ( B3
            = ( uminus_uminus_real @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_2975_eq__abs__iff_H,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( A2
        = ( abs_abs_Code_integer @ B3 ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
        & ( ( B3 = A2 )
          | ( B3
            = ( uminus1351360451143612070nteger @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_2976_eq__abs__iff_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( abs_abs_rat @ B3 ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
        & ( ( B3 = A2 )
          | ( B3
            = ( uminus_uminus_rat @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_2977_eq__abs__iff_H,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( abs_abs_int @ B3 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A2 )
        & ( ( B3 = A2 )
          | ( B3
            = ( uminus_uminus_int @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_2978_abs__eq__iff_H,axiom,
    ! [A2: real,B3: real] :
      ( ( ( abs_abs_real @ A2 )
        = B3 )
      = ( ( ord_less_eq_real @ zero_zero_real @ B3 )
        & ( ( A2 = B3 )
          | ( A2
            = ( uminus_uminus_real @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_2979_abs__eq__iff_H,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = B3 )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
        & ( ( A2 = B3 )
          | ( A2
            = ( uminus1351360451143612070nteger @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_2980_abs__eq__iff_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = B3 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
        & ( ( A2 = B3 )
          | ( A2
            = ( uminus_uminus_rat @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_2981_abs__eq__iff_H,axiom,
    ! [A2: int,B3: int] :
      ( ( ( abs_abs_int @ A2 )
        = B3 )
      = ( ( ord_less_eq_int @ zero_zero_int @ B3 )
        & ( ( A2 = B3 )
          | ( A2
            = ( uminus_uminus_int @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_2982_abs__minus__le__zero,axiom,
    ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A2 ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_2983_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A2 ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_2984_abs__minus__le__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A2 ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_2985_abs__minus__le__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A2 ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_2986_abs__div__pos,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y3 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_2987_abs__div__pos,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y3 )
        = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_2988_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_2989_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_2990_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_2991_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_2992_abs__of__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( abs_abs_int @ A2 )
        = ( uminus_uminus_int @ A2 ) ) ) ).

% abs_of_neg
thf(fact_2993_abs__of__neg,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( abs_abs_real @ A2 )
        = ( uminus_uminus_real @ A2 ) ) ) ).

% abs_of_neg
thf(fact_2994_abs__of__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A2 )
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% abs_of_neg
thf(fact_2995_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_neg
thf(fact_2996_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_2997_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_2998_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_2999_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_3000_length__removeAll__less__eq,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3001_length__removeAll__less__eq,axiom,
    ! [X2: $o,Xs: list_o] : ( ord_less_eq_nat @ ( size_size_list_o @ ( removeAll_o @ X2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3002_length__removeAll__less__eq,axiom,
    ! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_removeAll_less_eq
thf(fact_3003_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3004_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3005_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
           => ( P @ ( nth_o @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3006_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3007_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: real > $o,X2: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
         => ( P @ ( nth_real @ Xs @ I2 ) ) )
     => ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3008_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,X2: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
     => ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3009_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X2: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I2 ) ) )
     => ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3010_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3011_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: $o > $o,X2: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I2 ) ) )
     => ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3012_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X2: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3013_in__set__conv__nth,axiom,
    ! [X2: real,Xs: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3014_in__set__conv__nth,axiom,
    ! [X2: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3015_in__set__conv__nth,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3016_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3017_in__set__conv__nth,axiom,
    ! [X2: $o,Xs: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3018_in__set__conv__nth,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3019_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_3020_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_3021_list__ball__nth,axiom,
    ! [N: nat,Xs: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_3022_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_3023_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3024_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3025_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3026_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3027_nth__mem,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3028_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_3029_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).

% zabs_def
thf(fact_3030_length__removeAll__less,axiom,
    ! [X2: real,Xs: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_real @ ( removeAll_real @ X2 @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_3031_length__removeAll__less,axiom,
    ! [X2: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_3032_length__removeAll__less,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_3033_length__removeAll__less,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_3034_length__removeAll__less,axiom,
    ! [X2: $o,Xs: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_o @ ( removeAll_o @ X2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_3035_length__removeAll__less,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_removeAll_less
thf(fact_3036_power__diff__power__eq,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
            = ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3037_power__diff__power__eq,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
            = ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3038_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_3039_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3040_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3041_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3042_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3043_power__decreasing__iff,axiom,
    ! [B3: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( ord_less_real @ B3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3044_power__decreasing__iff,axiom,
    ! [B3: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B3 )
     => ( ( ord_less_rat @ B3 @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3045_power__decreasing__iff,axiom,
    ! [B3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ B3 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3046_power__decreasing__iff,axiom,
    ! [B3: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ B3 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_3047_zero__less__power__abs__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) )
      = ( ( A2 != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_3048_zero__less__power__abs__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
      = ( ( A2 != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_3049_zero__less__power__abs__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) )
      = ( ( A2 != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_3050_zero__less__power__abs__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) )
      = ( ( A2 != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_3051_power__mono__iff,axiom,
    ! [A2: real,B3: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) )
            = ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_3052_power__mono__iff,axiom,
    ! [A2: rat,B3: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) )
            = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_3053_power__mono__iff,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
            = ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_3054_power__mono__iff,axiom,
    ! [A2: int,B3: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) )
            = ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_3055_power__increasing__iff,axiom,
    ! [B3: real,X2: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y3 ) )
        = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_3056_power__increasing__iff,axiom,
    ! [B3: rat,X2: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y3 ) )
        = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_3057_power__increasing__iff,axiom,
    ! [B3: nat,X2: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B3 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y3 ) )
        = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_3058_power__increasing__iff,axiom,
    ! [B3: int,X2: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y3 ) )
        = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_3059_power__strict__decreasing__iff,axiom,
    ! [B3: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( ord_less_real @ B3 @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3060_power__strict__decreasing__iff,axiom,
    ! [B3: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B3 )
     => ( ( ord_less_rat @ B3 @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3061_power__strict__decreasing__iff,axiom,
    ! [B3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ B3 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3062_power__strict__decreasing__iff,axiom,
    ! [B3: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ B3 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3063_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3064_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3065_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3066_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3067_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3068_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3069_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3070_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3071_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3072_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3073_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3074_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3075_even__odd__cases,axiom,
    ! [X2: nat] :
      ( ! [N3: nat] :
          ( X2
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X2
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_3076_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_3077_power__shift,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ( power_power_nat @ X2 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_3078_add__left__cancel,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ( plus_plus_real @ A2 @ B3 )
        = ( plus_plus_real @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add_left_cancel
thf(fact_3079_add__left__cancel,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B3 )
        = ( plus_plus_rat @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add_left_cancel
thf(fact_3080_add__left__cancel,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B3 )
        = ( plus_plus_nat @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add_left_cancel
thf(fact_3081_add__left__cancel,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B3 )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add_left_cancel
thf(fact_3082_add__right__cancel,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ( plus_plus_real @ B3 @ A2 )
        = ( plus_plus_real @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add_right_cancel
thf(fact_3083_add__right__cancel,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B3 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add_right_cancel
thf(fact_3084_add__right__cancel,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B3 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add_right_cancel
thf(fact_3085_add__right__cancel,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B3 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add_right_cancel
thf(fact_3086_add__le__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
      = ( ord_less_eq_real @ A2 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_3087_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
      = ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_3088_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
      = ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_3089_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
      = ( ord_less_eq_int @ A2 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_3090_add__le__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
      = ( ord_less_eq_real @ A2 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_3091_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
      = ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_3092_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
      = ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_3093_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
      = ( ord_less_eq_int @ A2 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_3094_double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( plus_plus_real @ A2 @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_3095_double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( plus_plus_rat @ A2 @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_3096_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( plus_plus_int @ A2 @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_3097_add__0,axiom,
    ! [A2: literal] :
      ( ( plus_plus_literal @ zero_zero_literal @ A2 )
      = A2 ) ).

% add_0
thf(fact_3098_add__0,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% add_0
thf(fact_3099_add__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add_0
thf(fact_3100_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_3101_add__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add_0
thf(fact_3102_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y3 ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_3103_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y3 )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_3104_add__cancel__right__right,axiom,
    ! [A2: real,B3: real] :
      ( ( A2
        = ( plus_plus_real @ A2 @ B3 ) )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_3105_add__cancel__right__right,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( plus_plus_rat @ A2 @ B3 ) )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_3106_add__cancel__right__right,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B3 ) )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_3107_add__cancel__right__right,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( plus_plus_int @ A2 @ B3 ) )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_3108_add__cancel__right__left,axiom,
    ! [A2: real,B3: real] :
      ( ( A2
        = ( plus_plus_real @ B3 @ A2 ) )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_3109_add__cancel__right__left,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( plus_plus_rat @ B3 @ A2 ) )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_3110_add__cancel__right__left,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2
        = ( plus_plus_nat @ B3 @ A2 ) )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_3111_add__cancel__right__left,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( plus_plus_int @ B3 @ A2 ) )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_3112_add__cancel__left__right,axiom,
    ! [A2: real,B3: real] :
      ( ( ( plus_plus_real @ A2 @ B3 )
        = A2 )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_3113_add__cancel__left__right,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( plus_plus_rat @ A2 @ B3 )
        = A2 )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_3114_add__cancel__left__right,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( plus_plus_nat @ A2 @ B3 )
        = A2 )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_3115_add__cancel__left__right,axiom,
    ! [A2: int,B3: int] :
      ( ( ( plus_plus_int @ A2 @ B3 )
        = A2 )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_3116_add__cancel__left__left,axiom,
    ! [B3: real,A2: real] :
      ( ( ( plus_plus_real @ B3 @ A2 )
        = A2 )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_3117_add__cancel__left__left,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ( plus_plus_rat @ B3 @ A2 )
        = A2 )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_3118_add__cancel__left__left,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B3 @ A2 )
        = A2 )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_3119_add__cancel__left__left,axiom,
    ! [B3: int,A2: int] :
      ( ( ( plus_plus_int @ B3 @ A2 )
        = A2 )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_3120_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A2 @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_3121_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A2 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_3122_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_3123_add_Oright__neutral,axiom,
    ! [A2: literal] :
      ( ( plus_plus_literal @ A2 @ zero_zero_literal )
      = A2 ) ).

% add.right_neutral
thf(fact_3124_add_Oright__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% add.right_neutral
thf(fact_3125_add_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.right_neutral
thf(fact_3126_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_3127_add_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.right_neutral
thf(fact_3128_add__less__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
      = ( ord_less_real @ A2 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_3129_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
      = ( ord_less_rat @ A2 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_3130_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
      = ( ord_less_nat @ A2 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_3131_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
      = ( ord_less_int @ A2 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_3132_add__less__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
      = ( ord_less_real @ A2 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_3133_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
      = ( ord_less_rat @ A2 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_3134_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
      = ( ord_less_nat @ A2 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_3135_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
      = ( ord_less_int @ A2 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_3136_add__diff__cancel,axiom,
    ! [A2: real,B3: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3137_add__diff__cancel,axiom,
    ! [A2: rat,B3: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3138_add__diff__cancel,axiom,
    ! [A2: int,B3: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3139_diff__add__cancel,axiom,
    ! [A2: real,B3: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3140_diff__add__cancel,axiom,
    ! [A2: rat,B3: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3141_diff__add__cancel,axiom,
    ! [A2: int,B3: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3142_add__diff__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
      = ( minus_minus_real @ A2 @ B3 ) ) ).

% add_diff_cancel_left
thf(fact_3143_add__diff__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
      = ( minus_minus_rat @ A2 @ B3 ) ) ).

% add_diff_cancel_left
thf(fact_3144_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
      = ( minus_minus_nat @ A2 @ B3 ) ) ).

% add_diff_cancel_left
thf(fact_3145_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
      = ( minus_minus_int @ A2 @ B3 ) ) ).

% add_diff_cancel_left
thf(fact_3146_add__diff__cancel__left_H,axiom,
    ! [A2: real,B3: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ A2 )
      = B3 ) ).

% add_diff_cancel_left'
thf(fact_3147_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ A2 )
      = B3 ) ).

% add_diff_cancel_left'
thf(fact_3148_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ A2 )
      = B3 ) ).

% add_diff_cancel_left'
thf(fact_3149_add__diff__cancel__left_H,axiom,
    ! [A2: int,B3: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ A2 )
      = B3 ) ).

% add_diff_cancel_left'
thf(fact_3150_add__diff__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
      = ( minus_minus_real @ A2 @ B3 ) ) ).

% add_diff_cancel_right
thf(fact_3151_add__diff__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ A2 @ B3 ) ) ).

% add_diff_cancel_right
thf(fact_3152_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
      = ( minus_minus_nat @ A2 @ B3 ) ) ).

% add_diff_cancel_right
thf(fact_3153_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
      = ( minus_minus_int @ A2 @ B3 ) ) ).

% add_diff_cancel_right
thf(fact_3154_add__diff__cancel__right_H,axiom,
    ! [A2: real,B3: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3155_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B3: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3156_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3157_add__diff__cancel__right_H,axiom,
    ! [A2: int,B3: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_3158_add__minus__cancel,axiom,
    ! [A2: int,B3: int] :
      ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) )
      = B3 ) ).

% add_minus_cancel
thf(fact_3159_add__minus__cancel,axiom,
    ! [A2: real,B3: real] :
      ( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B3 ) )
      = B3 ) ).

% add_minus_cancel
thf(fact_3160_add__minus__cancel,axiom,
    ! [A2: rat,B3: rat] :
      ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) )
      = B3 ) ).

% add_minus_cancel
thf(fact_3161_add__minus__cancel,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) )
      = B3 ) ).

% add_minus_cancel
thf(fact_3162_add__minus__cancel,axiom,
    ! [A2: complex,B3: complex] :
      ( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) )
      = B3 ) ).

% add_minus_cancel
thf(fact_3163_minus__add__cancel,axiom,
    ! [A2: int,B3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B3 ) )
      = B3 ) ).

% minus_add_cancel
thf(fact_3164_minus__add__cancel,axiom,
    ! [A2: real,B3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B3 ) )
      = B3 ) ).

% minus_add_cancel
thf(fact_3165_minus__add__cancel,axiom,
    ! [A2: rat,B3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B3 ) )
      = B3 ) ).

% minus_add_cancel
thf(fact_3166_minus__add__cancel,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
      = B3 ) ).

% minus_add_cancel
thf(fact_3167_minus__add__cancel,axiom,
    ! [A2: complex,B3: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( plus_plus_complex @ A2 @ B3 ) )
      = B3 ) ).

% minus_add_cancel
thf(fact_3168_minus__add__distrib,axiom,
    ! [A2: int,B3: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B3 ) ) ) ).

% minus_add_distrib
thf(fact_3169_minus__add__distrib,axiom,
    ! [A2: real,B3: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B3 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B3 ) ) ) ).

% minus_add_distrib
thf(fact_3170_minus__add__distrib,axiom,
    ! [A2: rat,B3: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B3 ) ) ) ).

% minus_add_distrib
thf(fact_3171_minus__add__distrib,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B3 ) ) ) ).

% minus_add_distrib
thf(fact_3172_minus__add__distrib,axiom,
    ! [A2: complex,B3: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).

% minus_add_distrib
thf(fact_3173_abs__add__abs,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).

% abs_add_abs
thf(fact_3174_abs__add__abs,axiom,
    ! [A2: real,B3: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_add_abs
thf(fact_3175_abs__add__abs,axiom,
    ! [A2: rat,B3: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_add_abs
thf(fact_3176_abs__add__abs,axiom,
    ! [A2: int,B3: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).

% abs_add_abs
thf(fact_3177_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_3178_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_3179_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_3180_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_3181_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_3182_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_3183_add__le__same__cancel1,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A2 ) @ B3 )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_3184_add__le__same__cancel1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B3 @ A2 ) @ B3 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_3185_add__le__same__cancel1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_3186_add__le__same__cancel1,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_3187_add__le__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_3188_add__le__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_3189_add__le__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_3190_add__le__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_3191_le__add__same__cancel1,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_3192_le__add__same__cancel1,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_3193_le__add__same__cancel1,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_3194_le__add__same__cancel1,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_3195_le__add__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B3 @ A2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_3196_le__add__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B3 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_3197_le__add__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_3198_le__add__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_3199_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3200_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3201_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3202_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3203_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3204_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3205_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3206_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3207_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3208_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3209_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3210_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3211_less__add__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ ( plus_plus_real @ B3 @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_3212_less__add__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B3 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_3213_less__add__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_3214_less__add__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_3215_less__add__same__cancel1,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B3 ) )
      = ( ord_less_real @ zero_zero_real @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_3216_less__add__same__cancel1,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B3 ) )
      = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_3217_less__add__same__cancel1,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
      = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_3218_less__add__same__cancel1,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
      = ( ord_less_int @ zero_zero_int @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_3219_add__less__same__cancel2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_3220_add__less__same__cancel2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_3221_add__less__same__cancel2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_3222_add__less__same__cancel2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_3223_add__less__same__cancel1,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B3 @ A2 ) @ B3 )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_3224_add__less__same__cancel1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B3 @ A2 ) @ B3 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_3225_add__less__same__cancel1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_3226_add__less__same__cancel1,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_3227_le__add__diff__inverse2,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3228_le__add__diff__inverse2,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3229_le__add__diff__inverse2,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3230_le__add__diff__inverse2,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_3231_le__add__diff__inverse,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A2 @ B3 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3232_le__add__diff__inverse,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A2 @ B3 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3233_le__add__diff__inverse,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3234_le__add__diff__inverse,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_3235_diff__add__zero,axiom,
    ! [A2: nat,B3: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_3236_add_Oright__inverse,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_3237_add_Oright__inverse,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_3238_add_Oright__inverse,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_3239_add_Oright__inverse,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_3240_add_Oright__inverse,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ A2 ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_3241_ab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_3242_ab__left__minus,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_3243_ab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_3244_ab__left__minus,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_3245_ab__left__minus,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_3246_power__inject__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ( power_power_real @ A2 @ M )
          = ( power_power_real @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3247_power__inject__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ( power_power_rat @ A2 @ M )
          = ( power_power_rat @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3248_power__inject__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ( power_power_nat @ A2 @ M )
          = ( power_power_nat @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3249_power__inject__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( power_power_int @ A2 @ M )
          = ( power_power_int @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_3250_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_3251_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_3252_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_3253_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_3254_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_3255_diff__minus__eq__add,axiom,
    ! [A2: int,B3: int] :
      ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B3 ) )
      = ( plus_plus_int @ A2 @ B3 ) ) ).

% diff_minus_eq_add
thf(fact_3256_diff__minus__eq__add,axiom,
    ! [A2: real,B3: real] :
      ( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B3 ) )
      = ( plus_plus_real @ A2 @ B3 ) ) ).

% diff_minus_eq_add
thf(fact_3257_diff__minus__eq__add,axiom,
    ! [A2: rat,B3: rat] :
      ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
      = ( plus_plus_rat @ A2 @ B3 ) ) ).

% diff_minus_eq_add
thf(fact_3258_diff__minus__eq__add,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
      = ( plus_p5714425477246183910nteger @ A2 @ B3 ) ) ).

% diff_minus_eq_add
thf(fact_3259_diff__minus__eq__add,axiom,
    ! [A2: complex,B3: complex] :
      ( ( minus_minus_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) )
      = ( plus_plus_complex @ A2 @ B3 ) ) ).

% diff_minus_eq_add
thf(fact_3260_uminus__add__conv__diff,axiom,
    ! [A2: int,B3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 )
      = ( minus_minus_int @ B3 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_3261_uminus__add__conv__diff,axiom,
    ! [A2: real,B3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B3 )
      = ( minus_minus_real @ B3 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_3262_uminus__add__conv__diff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
      = ( minus_minus_rat @ B3 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_3263_uminus__add__conv__diff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
      = ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_3264_uminus__add__conv__diff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
      = ( minus_minus_complex @ B3 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_3265_power__Suc0__right,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_3266_power__Suc0__right,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_3267_power__Suc0__right,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_3268_power__Suc0__right,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_3269_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_3270_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_3271_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_3272_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_3273_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_3274_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_3275_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M: nat] :
      ( ( ( power_power_nat @ X2 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_3276_nat__zero__less__power__iff,axiom,
    ! [X2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_3277_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_3278_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3279_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3280_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_3281_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_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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_3289_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_3290_power__strict__increasing__iff,axiom,
    ! [B3: real,X2: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y3 ) )
        = ( ord_less_nat @ X2 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_3291_power__strict__increasing__iff,axiom,
    ! [B3: rat,X2: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B3 )
     => ( ( ord_less_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y3 ) )
        = ( ord_less_nat @ X2 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_3292_power__strict__increasing__iff,axiom,
    ! [B3: nat,X2: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B3 )
     => ( ( ord_less_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y3 ) )
        = ( ord_less_nat @ X2 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_3293_power__strict__increasing__iff,axiom,
    ! [B3: int,X2: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B3 )
     => ( ( ord_less_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y3 ) )
        = ( ord_less_nat @ X2 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_3294_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( ( power_power_rat @ A2 @ N )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3295_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( ( power_power_int @ A2 @ N )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3296_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( ( power_power_nat @ A2 @ N )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3297_power__eq__0__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( ( power_power_real @ A2 @ N )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3298_power__eq__0__iff,axiom,
    ! [A2: complex,N: nat] :
      ( ( ( power_power_complex @ A2 @ N )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_3299_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_3300_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3301_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3302_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3303_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_3304_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_3305_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3306_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3307_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3308_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3309_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W2: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B3 ) @ C )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3311_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3312_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3313_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_3314_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3315_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3316_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3317_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3318_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A2: real,B3: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A2 ) )
     => ( ( plus_plus_real @ A3 @ B3 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).

% group_cancel.add1
thf(fact_3319_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A2: rat,B3: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( plus_plus_rat @ A3 @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).

% group_cancel.add1
thf(fact_3320_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A2: nat,B3: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( plus_plus_nat @ A3 @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).

% group_cancel.add1
thf(fact_3321_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A2: int,B3: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( plus_plus_int @ A3 @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).

% group_cancel.add1
thf(fact_3322_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B3: real,A2: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B3 ) )
     => ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).

% group_cancel.add2
thf(fact_3323_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B3: rat,A2: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B3 ) )
     => ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).

% group_cancel.add2
thf(fact_3324_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B3: nat,A2: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B3 ) )
     => ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).

% group_cancel.add2
thf(fact_3325_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B3: int,A2: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B3 ) )
     => ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).

% group_cancel.add2
thf(fact_3326_add_Oassoc,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B3 ) @ C )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).

% add.assoc
thf(fact_3327_add_Oassoc,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% add.assoc
thf(fact_3328_add_Oassoc,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% add.assoc
thf(fact_3329_add_Oassoc,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).

% add.assoc
thf(fact_3330_add_Oleft__cancel,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ( plus_plus_real @ A2 @ B3 )
        = ( plus_plus_real @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add.left_cancel
thf(fact_3331_add_Oleft__cancel,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B3 )
        = ( plus_plus_rat @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add.left_cancel
thf(fact_3332_add_Oleft__cancel,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B3 )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B3 = C ) ) ).

% add.left_cancel
thf(fact_3333_add_Oright__cancel,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ( plus_plus_real @ B3 @ A2 )
        = ( plus_plus_real @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add.right_cancel
thf(fact_3334_add_Oright__cancel,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B3 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add.right_cancel
thf(fact_3335_add_Oright__cancel,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B3 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B3 = C ) ) ).

% add.right_cancel
thf(fact_3336_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_3337_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_3338_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_3339_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).

% add.commute
thf(fact_3340_add_Oleft__commute,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( plus_plus_real @ B3 @ ( plus_plus_real @ A2 @ C ) )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).

% add.left_commute
thf(fact_3341_add_Oleft__commute,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( plus_plus_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% add.left_commute
thf(fact_3342_add_Oleft__commute,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% add.left_commute
thf(fact_3343_add_Oleft__commute,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( plus_plus_int @ B3 @ ( plus_plus_int @ A2 @ C ) )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).

% add.left_commute
thf(fact_3344_add__left__imp__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ( plus_plus_real @ A2 @ B3 )
        = ( plus_plus_real @ A2 @ C ) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
thf(fact_3345_add__left__imp__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B3 )
        = ( plus_plus_rat @ A2 @ C ) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
thf(fact_3346_add__left__imp__eq,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B3 )
        = ( plus_plus_nat @ A2 @ C ) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
thf(fact_3347_add__left__imp__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B3 )
        = ( plus_plus_int @ A2 @ C ) )
     => ( B3 = C ) ) ).

% add_left_imp_eq
thf(fact_3348_add__right__imp__eq,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ( plus_plus_real @ B3 @ A2 )
        = ( plus_plus_real @ C @ A2 ) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
thf(fact_3349_add__right__imp__eq,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B3 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
thf(fact_3350_add__right__imp__eq,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B3 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
thf(fact_3351_add__right__imp__eq,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B3 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
     => ( B3 = C ) ) ).

% add_right_imp_eq
thf(fact_3352_add__le__imp__le__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
     => ( ord_less_eq_real @ A2 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_3353_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
     => ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_3354_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
     => ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_3355_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
     => ( ord_less_eq_int @ A2 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_3356_add__le__imp__le__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
     => ( ord_less_eq_real @ A2 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_3357_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
     => ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_3358_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
     => ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_3359_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
     => ( ord_less_eq_int @ A2 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_3360_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C5: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_3361_add__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_3362_add__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_3363_add__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_3364_add__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_3365_less__eqE,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ~ ! [C3: nat] :
            ( B3
           != ( plus_plus_nat @ A2 @ C3 ) ) ) ).

% less_eqE
thf(fact_3366_add__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_3367_add__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_3368_add__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_3369_add__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_3370_add__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_3371_add__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_3372_add__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_3373_add__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_3374_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3375_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3376_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3377_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3378_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3379_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3380_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3381_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3382_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3383_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3384_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3385_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3386_verit__sum__simplify,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% verit_sum_simplify
thf(fact_3387_verit__sum__simplify,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_3388_verit__sum__simplify,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_3389_verit__sum__simplify,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% verit_sum_simplify
thf(fact_3390_add_Ogroup__left__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_3391_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_3392_add_Ogroup__left__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_3393_add_Ocomm__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% add.comm_neutral
thf(fact_3394_add_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.comm_neutral
thf(fact_3395_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_3396_add_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.comm_neutral
thf(fact_3397_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3398_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3399_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3400_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_3401_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3402_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3403_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3404_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3405_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3406_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3407_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3408_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3409_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3410_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3411_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3412_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3413_add__strict__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3414_add__strict__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3415_add__strict__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3416_add__strict__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_3417_add__strict__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_3418_add__strict__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_3419_add__strict__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_3420_add__strict__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_3421_add__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3422_add__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3423_add__strict__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3424_add__strict__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_3425_add__less__imp__less__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
     => ( ord_less_real @ A2 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_3426_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
     => ( ord_less_rat @ A2 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_3427_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
     => ( ord_less_nat @ A2 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_3428_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
     => ( ord_less_int @ A2 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_3429_add__less__imp__less__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
     => ( ord_less_real @ A2 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_3430_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
     => ( ord_less_rat @ A2 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_3431_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
     => ( ord_less_nat @ A2 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_3432_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
     => ( ord_less_int @ A2 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_3433_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A2: real,B3: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A2 ) )
     => ( ( minus_minus_real @ A3 @ B3 )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3434_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A2: rat,B3: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( minus_minus_rat @ A3 @ B3 )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3435_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A2: int,B3: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( minus_minus_int @ A3 @ B3 )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub1
thf(fact_3436_diff__eq__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ( minus_minus_real @ A2 @ B3 )
        = C )
      = ( A2
        = ( plus_plus_real @ C @ B3 ) ) ) ).

% diff_eq_eq
thf(fact_3437_diff__eq__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ( minus_minus_rat @ A2 @ B3 )
        = C )
      = ( A2
        = ( plus_plus_rat @ C @ B3 ) ) ) ).

% diff_eq_eq
thf(fact_3438_diff__eq__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ( minus_minus_int @ A2 @ B3 )
        = C )
      = ( A2
        = ( plus_plus_int @ C @ B3 ) ) ) ).

% diff_eq_eq
thf(fact_3439_eq__diff__eq,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( A2
        = ( minus_minus_real @ C @ B3 ) )
      = ( ( plus_plus_real @ A2 @ B3 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3440_eq__diff__eq,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( A2
        = ( minus_minus_rat @ C @ B3 ) )
      = ( ( plus_plus_rat @ A2 @ B3 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3441_eq__diff__eq,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( A2
        = ( minus_minus_int @ C @ B3 ) )
      = ( ( plus_plus_int @ A2 @ B3 )
        = C ) ) ).

% eq_diff_eq
thf(fact_3442_add__diff__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( plus_plus_real @ A2 @ ( minus_minus_real @ B3 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).

% add_diff_eq
thf(fact_3443_add__diff__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).

% add_diff_eq
thf(fact_3444_add__diff__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).

% add_diff_eq
thf(fact_3445_diff__diff__eq2,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( minus_minus_real @ A2 @ ( minus_minus_real @ B3 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ).

% diff_diff_eq2
thf(fact_3446_diff__diff__eq2,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ).

% diff_diff_eq2
thf(fact_3447_diff__diff__eq2,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).

% diff_diff_eq2
thf(fact_3448_diff__add__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ).

% diff_add_eq
thf(fact_3449_diff__add__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ).

% diff_add_eq
thf(fact_3450_diff__add__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).

% diff_add_eq
thf(fact_3451_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( minus_minus_real @ A2 @ ( plus_plus_real @ B3 @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B3 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3452_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B3 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3453_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3454_add__implies__diff,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ( plus_plus_real @ C @ B3 )
        = A2 )
     => ( C
        = ( minus_minus_real @ A2 @ B3 ) ) ) ).

% add_implies_diff
thf(fact_3455_add__implies__diff,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ( plus_plus_rat @ C @ B3 )
        = A2 )
     => ( C
        = ( minus_minus_rat @ A2 @ B3 ) ) ) ).

% add_implies_diff
thf(fact_3456_add__implies__diff,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C @ B3 )
        = A2 )
     => ( C
        = ( minus_minus_nat @ A2 @ B3 ) ) ) ).

% add_implies_diff
thf(fact_3457_add__implies__diff,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( ( plus_plus_int @ C @ B3 )
        = A2 )
     => ( C
        = ( minus_minus_int @ A2 @ B3 ) ) ) ).

% add_implies_diff
thf(fact_3458_diff__diff__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
      = ( minus_minus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3459_diff__diff__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
      = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3460_diff__diff__eq,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3461_diff__diff__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
      = ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).

% diff_diff_eq
thf(fact_3462_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A2: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( uminus_uminus_int @ A3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3463_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A2: real] :
      ( ( A3
        = ( plus_plus_real @ K @ A2 ) )
     => ( ( uminus_uminus_real @ A3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3464_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A2: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( uminus_uminus_rat @ A3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3465_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A2: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A2 ) )
     => ( ( uminus1351360451143612070nteger @ A3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3466_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A2: complex] :
      ( ( A3
        = ( plus_plus_complex @ K @ A2 ) )
     => ( ( uminus1482373934393186551omplex @ A3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_3467_add_Oinverse__distrib__swap,axiom,
    ! [A2: int,B3: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3468_add_Oinverse__distrib__swap,axiom,
    ! [A2: real,B3: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B3 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3469_add_Oinverse__distrib__swap,axiom,
    ! [A2: rat,B3: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3470_add_Oinverse__distrib__swap,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3471_add_Oinverse__distrib__swap,axiom,
    ! [A2: complex,B3: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_3472_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( suc @ A3 )
        = ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).

% nat_arith.suc1
thf(fact_3473_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_3474_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_3475_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_3476_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_3477_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_3478_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_3479_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_3480_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_3481_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_3482_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_3483_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_3484_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_3485_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_3486_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_3487_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_3488_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_3489_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_3490_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_3491_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_3492_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_3493_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_3494_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_3495_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_3496_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_3497_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_3498_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_3499_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_3500_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_3501_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_real_def
thf(fact_3502_add__decreasing,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B3 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_3503_add__decreasing,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B3 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_3504_add__decreasing,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_3505_add__decreasing,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B3 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_3506_add__increasing,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_3507_add__increasing,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_3508_add__increasing,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_3509_add__increasing,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_3510_add__decreasing2,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ B3 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_3511_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ B3 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_3512_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_3513_add__decreasing2,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A2 @ B3 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_3514_add__increasing2,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B3 @ A2 )
       => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_3515_add__increasing2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B3 @ A2 )
       => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_3516_add__increasing2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B3 @ A2 )
       => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_3517_add__increasing2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B3 @ A2 )
       => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_3518_add__nonneg__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3519_add__nonneg__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3520_add__nonneg__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3521_add__nonneg__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3522_add__nonpos__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_3523_add__nonpos__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3524_add__nonpos__nonpos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3525_add__nonpos__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3526_add__nonneg__eq__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ X2 @ Y3 )
            = zero_zero_real )
          = ( ( X2 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3527_add__nonneg__eq__0__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( plus_plus_rat @ X2 @ Y3 )
            = zero_zero_rat )
          = ( ( X2 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3528_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X2 @ Y3 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3529_add__nonneg__eq__0__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( plus_plus_int @ X2 @ Y3 )
            = zero_zero_int )
          = ( ( X2 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3530_add__nonpos__eq__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X2 @ Y3 )
            = zero_zero_real )
          = ( ( X2 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3531_add__nonpos__eq__0__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X2 @ Y3 )
            = zero_zero_rat )
          = ( ( X2 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3532_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X2 @ Y3 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3533_add__nonpos__eq__0__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X2 @ Y3 )
            = zero_zero_int )
          = ( ( X2 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3534_add__less__le__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3535_add__less__le__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3536_add__less__le__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3537_add__less__le__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_3538_add__le__less__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3539_add__le__less__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3540_add__le__less__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3541_add__le__less__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_3542_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3543_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3544_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3545_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3546_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3547_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3548_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3549_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3550_pos__add__strict,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3551_pos__add__strict,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3552_pos__add__strict,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3553_pos__add__strict,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3554_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ~ ! [C3: nat] :
            ( ( B3
              = ( plus_plus_nat @ A2 @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3555_add__pos__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_3556_add__pos__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_3557_add__pos__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_3558_add__pos__pos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_3559_add__neg__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_3560_add__neg__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_3561_add__neg__neg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3562_add__neg__neg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3563_add__less__zeroD,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
        | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_3564_add__less__zeroD,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X2 @ zero_zero_rat )
        | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_3565_add__less__zeroD,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
     => ( ( ord_less_int @ X2 @ zero_zero_int )
        | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_3566_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_3567_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( ( ( minus_minus_nat @ B3 @ A2 )
            = C )
          = ( B3
            = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3568_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B3 @ A2 ) )
        = B3 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3569_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3570_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3571_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3572_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3573_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3574_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3575_le__add__diff,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_3576_diff__add,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ A2 )
        = B3 ) ) ).

% diff_add
thf(fact_3577_le__diff__eq,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C @ B3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).

% le_diff_eq
thf(fact_3578_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).

% le_diff_eq
thf(fact_3579_le__diff__eq,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).

% le_diff_eq
thf(fact_3580_diff__le__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
      = ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C @ B3 ) ) ) ).

% diff_le_eq
thf(fact_3581_diff__le__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
      = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% diff_le_eq
thf(fact_3582_diff__le__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
      = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).

% diff_le_eq
thf(fact_3583_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3584_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3585_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3586_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3587_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3588_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3589_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3590_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3591_less__add__one,axiom,
    ! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).

% less_add_one
thf(fact_3592_less__add__one,axiom,
    ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).

% less_add_one
thf(fact_3593_less__add__one,axiom,
    ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).

% less_add_one
thf(fact_3594_less__add__one,axiom,
    ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).

% less_add_one
thf(fact_3595_add__mono1,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).

% add_mono1
thf(fact_3596_add__mono1,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B3 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_3597_add__mono1,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3598_add__mono1,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3599_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_3600_less__diff__eq,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ A2 @ ( minus_minus_real @ C @ B3 ) )
      = ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).

% less_diff_eq
thf(fact_3601_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B3 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).

% less_diff_eq
thf(fact_3602_less__diff__eq,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
      = ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).

% less_diff_eq
thf(fact_3603_diff__less__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
      = ( ord_less_real @ A2 @ ( plus_plus_real @ C @ B3 ) ) ) ).

% diff_less_eq
thf(fact_3604_diff__less__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
      = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% diff_less_eq
thf(fact_3605_diff__less__eq,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
      = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).

% diff_less_eq
thf(fact_3606_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: real,B3: real] :
      ( ~ ( ord_less_real @ A2 @ B3 )
     => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A2 @ B3 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3607_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B3: rat] :
      ( ~ ( ord_less_rat @ A2 @ B3 )
     => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A2 @ B3 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3608_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B3: nat] :
      ( ~ ( ord_less_nat @ A2 @ B3 )
     => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3609_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B3: int] :
      ( ~ ( ord_less_int @ A2 @ B3 )
     => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3610_add__eq__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ( plus_plus_int @ A2 @ B3 )
        = zero_zero_int )
      = ( B3
        = ( uminus_uminus_int @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3611_add__eq__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( plus_plus_real @ A2 @ B3 )
        = zero_zero_real )
      = ( B3
        = ( uminus_uminus_real @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3612_add__eq__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( plus_plus_rat @ A2 @ B3 )
        = zero_zero_rat )
      = ( B3
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3613_add__eq__0__iff,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
        = zero_z3403309356797280102nteger )
      = ( B3
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3614_add__eq__0__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( plus_plus_complex @ A2 @ B3 )
        = zero_zero_complex )
      = ( B3
        = ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_3615_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3616_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3617_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3618_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3619_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_3620_add_Oinverse__unique,axiom,
    ! [A2: int,B3: int] :
      ( ( ( plus_plus_int @ A2 @ B3 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A2 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_3621_add_Oinverse__unique,axiom,
    ! [A2: real,B3: real] :
      ( ( ( plus_plus_real @ A2 @ B3 )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A2 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_3622_add_Oinverse__unique,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( plus_plus_rat @ A2 @ B3 )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A2 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_3623_add_Oinverse__unique,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_3624_add_Oinverse__unique,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( plus_plus_complex @ A2 @ B3 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_3625_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( uminus_uminus_int @ B3 ) )
      = ( ( plus_plus_int @ A2 @ B3 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3626_eq__neg__iff__add__eq__0,axiom,
    ! [A2: real,B3: real] :
      ( ( A2
        = ( uminus_uminus_real @ B3 ) )
      = ( ( plus_plus_real @ A2 @ B3 )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3627_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B3 ) )
      = ( ( plus_plus_rat @ A2 @ B3 )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3628_eq__neg__iff__add__eq__0,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ B3 ) )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3629_eq__neg__iff__add__eq__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ B3 ) )
      = ( ( plus_plus_complex @ A2 @ B3 )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_3630_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B3 )
      = ( ( plus_plus_int @ A2 @ B3 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3631_neg__eq__iff__add__eq__0,axiom,
    ! [A2: real,B3: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = B3 )
      = ( ( plus_plus_real @ A2 @ B3 )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3632_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B3 )
      = ( ( plus_plus_rat @ A2 @ B3 )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3633_neg__eq__iff__add__eq__0,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = B3 )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3634_neg__eq__iff__add__eq__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = B3 )
      = ( ( plus_plus_complex @ A2 @ B3 )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_3635_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B3: int,A2: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B3 ) )
     => ( ( minus_minus_int @ A2 @ B2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3636_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B3: real,A2: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B3 ) )
     => ( ( minus_minus_real @ A2 @ B2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3637_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B3: rat,A2: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B3 ) )
     => ( ( minus_minus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3638_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B3: code_integer,A2: code_integer] :
      ( ( B2
        = ( plus_p5714425477246183910nteger @ K @ B3 ) )
     => ( ( minus_8373710615458151222nteger @ A2 @ B2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3639_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B3: complex,A2: complex] :
      ( ( B2
        = ( plus_plus_complex @ K @ B3 ) )
     => ( ( minus_minus_complex @ A2 @ B2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A2 @ B3 ) ) ) ) ).

% group_cancel.sub2
thf(fact_3640_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3641_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3642_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3643_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3644_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3645_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3646_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3647_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3648_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3649_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3650_abs__triangle__ineq,axiom,
    ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_3651_abs__triangle__ineq,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_3652_abs__triangle__ineq,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A2 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_3653_abs__triangle__ineq,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_3654_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_3655_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_3656_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_3657_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_3658_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_3659_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_3660_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).

% less_natE
thf(fact_3661_real__arch__pow,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X2 @ N3 ) ) ) ).

% real_arch_pow
thf(fact_3662_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_3663_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_3664_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_3665_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_3666_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_3667_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_3668_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_3669_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_3670_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_3671_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_3672_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_3673_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_3674_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_3675_add__strict__increasing2,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3676_add__strict__increasing2,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3677_add__strict__increasing2,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3678_add__strict__increasing2,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3679_add__strict__increasing,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3680_add__strict__increasing,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3681_add__strict__increasing,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3682_add__strict__increasing,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3683_add__pos__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3684_add__pos__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3685_add__pos__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3686_add__pos__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_3687_add__nonpos__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_3688_add__nonpos__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3689_add__nonpos__neg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3690_add__nonpos__neg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3691_add__nonneg__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3692_add__nonneg__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3693_add__nonneg__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3694_add__nonneg__pos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_3695_add__neg__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_3696_add__neg__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3697_add__neg__nonpos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3698_add__neg__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3699_field__le__epsilon,axiom,
    ! [X2: real,Y3: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y3 @ E ) ) )
     => ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_3700_field__le__epsilon,axiom,
    ! [X2: rat,Y3: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y3 @ E ) ) )
     => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_3701_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3702_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3703_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3704_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3705_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3706_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3707_div__add__self2,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3708_div__add__self2,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3709_div__add__self1,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3710_div__add__self1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3711_gt__half__sum,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B3 ) ) ).

% gt_half_sum
thf(fact_3712_gt__half__sum,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B3 ) ) ).

% gt_half_sum
thf(fact_3713_less__half__sum,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ord_less_rat @ A2 @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_3714_less__half__sum,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ord_less_real @ A2 @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_3715_abs__diff__le__iff,axiom,
    ! [X2: code_integer,A2: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ R2 ) @ X2 )
        & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A2 @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_3716_abs__diff__le__iff,axiom,
    ! [X2: real,A2: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ R2 ) @ X2 )
        & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A2 @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_3717_abs__diff__le__iff,axiom,
    ! [X2: rat,A2: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ R2 ) @ X2 )
        & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A2 @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_3718_abs__diff__le__iff,axiom,
    ! [X2: int,A2: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R2 ) @ X2 )
        & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A2 @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_3719_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B3: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_3720_abs__diff__triangle__ineq,axiom,
    ! [A2: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_3721_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_3722_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B3: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_3723_abs__triangle__ineq4,axiom,
    ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_3724_abs__triangle__ineq4,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_3725_abs__triangle__ineq4,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_3726_abs__triangle__ineq4,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_3727_abs__diff__less__iff,axiom,
    ! [X2: code_integer,A2: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ R2 ) @ X2 )
        & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A2 @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_3728_abs__diff__less__iff,axiom,
    ! [X2: real,A2: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A2 @ R2 ) @ X2 )
        & ( ord_less_real @ X2 @ ( plus_plus_real @ A2 @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_3729_abs__diff__less__iff,axiom,
    ! [X2: rat,A2: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ R2 ) @ X2 )
        & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A2 @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_3730_abs__diff__less__iff,axiom,
    ! [X2: int,A2: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A2 ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A2 @ R2 ) @ X2 )
        & ( ord_less_int @ X2 @ ( plus_plus_int @ A2 @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_3731_real__arch__pow__inv,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N3 ) @ Y3 ) ) ) ).

% real_arch_pow_inv
thf(fact_3732_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B3: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B3 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B3 @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3733_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B3: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
      = ( ( ( ord_less_nat @ A2 @ B3 )
         => ( P @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A2
              = ( plus_plus_nat @ B3 @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_3734_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_3735_abs__add__one__gt__zero,axiom,
    ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_3736_abs__add__one__gt__zero,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_3737_abs__add__one__gt__zero,axiom,
    ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_3738_abs__add__one__gt__zero,axiom,
    ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_3739_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_3740_frac__add,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y3 ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y3 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_3741_frac__add,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X2 @ Y3 ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X2 @ Y3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_3742_option_Osize__gen_I2_J,axiom,
    ! [X2: nat > nat,X22: nat] :
      ( ( size_option_nat @ X2 @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3743_option_Osize__gen_I2_J,axiom,
    ! [X2: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X2 @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3744_option_Osize__gen_I2_J,axiom,
    ! [X2: num > nat,X22: num] :
      ( ( size_option_num @ X2 @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3745_power__not__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( A2 != zero_zero_rat )
     => ( ( power_power_rat @ A2 @ N )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_3746_power__not__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( power_power_int @ A2 @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_3747_power__not__zero,axiom,
    ! [A2: nat,N: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( power_power_nat @ A2 @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_3748_power__not__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( A2 != zero_zero_real )
     => ( ( power_power_real @ A2 @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_3749_power__not__zero,axiom,
    ! [A2: complex,N: nat] :
      ( ( A2 != zero_zero_complex )
     => ( ( power_power_complex @ A2 @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_3750_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_3751_power__mono,axiom,
    ! [A2: real,B3: real,N: nat] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ).

% power_mono
thf(fact_3752_power__mono,axiom,
    ! [A2: rat,B3: rat,N: nat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ).

% power_mono
thf(fact_3753_power__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ).

% power_mono
thf(fact_3754_power__mono,axiom,
    ! [A2: int,B3: int,N: nat] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ).

% power_mono
thf(fact_3755_zero__le__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3756_zero__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3757_zero__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3758_zero__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3759_zero__less__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_3760_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_3761_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_3762_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_3763_one__le__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A2 )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3764_one__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3765_one__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3766_one__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3767_power__0,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_3768_power__0,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3769_power__0,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3770_power__0,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_3771_power__0,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_3772_power__less__imp__less__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ A2 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_3773_power__less__imp__less__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ A2 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_3774_power__less__imp__less__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_3775_power__less__imp__less__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ A2 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_3776_power__le__one,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ A2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_3777_power__le__one,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_3778_power__le__one,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3779_power__le__one,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ A2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3780_power__inject__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( ( ( power_power_real @ A2 @ ( suc @ N ) )
        = ( power_power_real @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_3781_power__inject__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( ( ( power_power_rat @ A2 @ ( suc @ N ) )
        = ( power_power_rat @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_3782_power__inject__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( ( ( power_power_nat @ A2 @ ( suc @ N ) )
        = ( power_power_nat @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_3783_power__inject__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( ( ( power_power_int @ A2 @ ( suc @ N ) )
        = ( power_power_int @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( A2 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_3784_power__le__imp__le__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ ( power_power_real @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_3785_power__le__imp__le__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ ( power_power_rat @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_3786_power__le__imp__le__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ ( power_power_nat @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_3787_power__le__imp__le__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ ( power_power_int @ B3 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_3788_power__gt1,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3789_power__gt1,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3790_power__gt1,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3791_power__gt1,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3792_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_3793_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_3794_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_3795_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_3796_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_3797_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3798_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3799_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3800_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3801_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_real @ one_one_real @ A2 )
       => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_3802_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_rat @ one_one_rat @ A2 )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_3803_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_nat @ one_one_nat @ A2 )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_3804_power__strict__increasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_int @ one_one_int @ A2 )
       => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_3805_zero__le__power__abs,axiom,
    ! [A2: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_3806_zero__le__power__abs,axiom,
    ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_3807_zero__le__power__abs,axiom,
    ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_3808_zero__le__power__abs,axiom,
    ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_3809_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_real @ one_one_real @ A2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N6 ) ) ) ) ).

% power_increasing
thf(fact_3810_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N6 ) ) ) ) ).

% power_increasing
thf(fact_3811_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N6 ) ) ) ) ).

% power_increasing
thf(fact_3812_power__increasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_int @ one_one_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N6 ) ) ) ) ).

% power_increasing
thf(fact_3813_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_3814_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_3815_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_3816_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_3817_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_3818_power__Suc__le__self,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ A2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3819_power__Suc__le__self,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3820_power__Suc__le__self,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3821_power__Suc__le__self,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ A2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3822_power__Suc__less__one,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ A2 @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_3823_power__Suc__less__one,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_3824_power__Suc__less__one,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_3825_power__Suc__less__one,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_3826_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ( ord_less_real @ A2 @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A2 @ N6 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3827_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_rat @ A2 @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N6 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3828_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ A2 @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N6 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3829_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ( ord_less_int @ A2 @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A2 @ N6 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3830_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: real] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ A2 @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A2 @ N6 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3831_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N6 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3832_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N6 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3833_power__decreasing,axiom,
    ! [N: nat,N6: nat,A2: int] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ A2 @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A2 @ N6 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3834_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3835_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3836_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3837_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3838_power__eq__imp__eq__base,axiom,
    ! [A2: real,N: nat,B3: real] :
      ( ( ( power_power_real @ A2 @ N )
        = ( power_power_real @ B3 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3839_power__eq__imp__eq__base,axiom,
    ! [A2: rat,N: nat,B3: rat] :
      ( ( ( power_power_rat @ A2 @ N )
        = ( power_power_rat @ B3 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3840_power__eq__imp__eq__base,axiom,
    ! [A2: nat,N: nat,B3: nat] :
      ( ( ( power_power_nat @ A2 @ N )
        = ( power_power_nat @ B3 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3841_power__eq__imp__eq__base,axiom,
    ! [A2: int,N: nat,B3: int] :
      ( ( ( power_power_int @ A2 @ N )
        = ( power_power_int @ B3 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3842_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: real,B3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( ( ( power_power_real @ A2 @ N )
              = ( power_power_real @ B3 @ N ) )
            = ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3843_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: rat,B3: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( ( ( power_power_rat @ A2 @ N )
              = ( power_power_rat @ B3 @ N ) )
            = ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3844_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( ( ( power_power_nat @ A2 @ N )
              = ( power_power_nat @ B3 @ N ) )
            = ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3845_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: int,B3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( ( ( power_power_int @ A2 @ N )
              = ( power_power_int @ B3 @ N ) )
            = ( A2 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3846_self__le__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3847_self__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3848_self__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3849_self__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3850_one__less__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3851_one__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3852_one__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3853_one__less__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_3854_power__diff,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_3855_power__diff,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_3856_power__diff,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_3857_power__diff,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_3858_power__diff,axiom,
    ! [A2: complex,N: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_3859_power__strict__mono,axiom,
    ! [A2: real,B3: real,N: nat] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3860_power__strict__mono,axiom,
    ! [A2: rat,B3: rat,N: nat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3861_power__strict__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3862_power__strict__mono,axiom,
    ! [A2: int,B3: int,N: nat] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3863_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3864_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3865_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3866_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3867_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3868_add__shift,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_3869_lemma__interval,axiom,
    ! [A2: real,X2: real,B3: real] :
      ( ( ord_less_real @ A2 @ X2 )
     => ( ( ord_less_real @ X2 @ B3 )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D6 )
               => ( ( ord_less_eq_real @ A2 @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B3 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_3870_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N )
              = A2 )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A2 ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3871_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N )
              = A2 ) ) ) ) ).

% realpow_pos_nth
thf(fact_3872_lemma__interval__lt,axiom,
    ! [A2: real,X2: real,B3: real] :
      ( ( ord_less_real @ A2 @ X2 )
     => ( ( ord_less_real @ X2 @ B3 )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D6 )
               => ( ( ord_less_real @ A2 @ Y5 )
                  & ( ord_less_real @ Y5 @ B3 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_3873_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N ) )
            = A2 ) ) ) ).

% realpow_pos_nth2
thf(fact_3874_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_3875_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_3876_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_3877_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_3878_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_3879_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% zle_add1_eq_le
thf(fact_3880_real__0__less__add__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).

% real_0_less_add_iff
thf(fact_3881_real__add__less__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
      = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).

% real_add_less_0_iff
thf(fact_3882_real__add__le__0__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).

% real_add_le_0_iff
thf(fact_3883_real__0__le__add__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).

% real_0_le_add_iff
thf(fact_3884_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3885_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z )
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
thf(fact_3886_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3887_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z2: int] :
        ? [N2: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_3888_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z2: int] :
        ? [N2: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3889_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3890_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
      = ( ord_less_int @ W2 @ Z ) ) ).

% add1_zle_eq
thf(fact_3891_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ W2 @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_3892_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3893_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% nat_less_real_le
thf(fact_3894_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3895_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_3896_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_3897_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_3898_sin__bound__lemma,axiom,
    ! [X2: real,Y3: real,U: real,V: real] :
      ( ( X2 = Y3 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y3 ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_3899_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( P @ A @ B )
          = ( P @ B @ A ) )
     => ( ! [A: nat] : ( P @ A @ zero_zero_nat )
       => ( ! [A: nat,B: nat] :
              ( ( P @ A @ B )
             => ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
         => ( P @ A2 @ B3 ) ) ) ) ).

% Euclid_induct
thf(fact_3900_add__0__iff,axiom,
    ! [B3: real,A2: real] :
      ( ( B3
        = ( plus_plus_real @ B3 @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% add_0_iff
thf(fact_3901_add__0__iff,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3
        = ( plus_plus_rat @ B3 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_3902_add__0__iff,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3
        = ( plus_plus_nat @ B3 @ A2 ) )
      = ( A2 = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_3903_add__0__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( B3
        = ( plus_plus_int @ B3 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% add_0_iff
thf(fact_3904_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_eq_real @ X2 @ ( 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 @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_3905_frac__unique__iff,axiom,
    ! [X2: real,A2: real] :
      ( ( ( archim2898591450579166408c_real @ X2 )
        = A2 )
      = ( ( member_real @ ( minus_minus_real @ X2 @ A2 ) @ ring_1_Ints_real )
        & ( ord_less_eq_real @ zero_zero_real @ A2 )
        & ( ord_less_real @ A2 @ one_one_real ) ) ) ).

% frac_unique_iff
thf(fact_3906_frac__unique__iff,axiom,
    ! [X2: rat,A2: rat] :
      ( ( ( archimedean_frac_rat @ X2 )
        = A2 )
      = ( ( member_rat @ ( minus_minus_rat @ X2 @ A2 ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A2 )
        & ( ord_less_rat @ A2 @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_3907_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3908_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
       => ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) @ I )
          = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3909_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys2 ) )
       => ( ( nth_Pr7514405829937366042_int_o @ ( zip_int_o @ Xs @ Ys2 ) @ I )
          = ( product_Pair_int_o @ ( nth_int @ Xs @ I ) @ ( nth_o @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3910_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) @ I )
          = ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3911_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) @ I )
          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3912_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
       => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) @ I )
          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3913_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys2 ) )
       => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys2 ) @ I )
          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3914_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
       => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) @ I )
          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3915_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys2: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
       => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys2 ) @ I )
          = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3916_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
       => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys2 ) @ I )
          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).

% nth_zip
thf(fact_3917_exp__less__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ).

% exp_less_mono
thf(fact_3918_exp__less__cancel__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_real @ X2 @ Y3 ) ) ).

% exp_less_cancel_iff
thf(fact_3919_exp__le__cancel__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% exp_le_cancel_iff
thf(fact_3920_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_3921_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_3922_frac__eq__0__iff,axiom,
    ! [X2: real] :
      ( ( ( archim2898591450579166408c_real @ X2 )
        = zero_zero_real )
      = ( member_real @ X2 @ ring_1_Ints_real ) ) ).

% frac_eq_0_iff
thf(fact_3923_frac__eq__0__iff,axiom,
    ! [X2: rat] :
      ( ( ( archimedean_frac_rat @ X2 )
        = zero_zero_rat )
      = ( member_rat @ X2 @ ring_1_Ints_rat ) ) ).

% frac_eq_0_iff
thf(fact_3924_exp__less__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_3925_one__less__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% one_less_exp_iff
thf(fact_3926_exp__le__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_3927_one__le__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% one_le_exp_iff
thf(fact_3928_frac__gt__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) )
      = ( ~ ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).

% frac_gt_0_iff
thf(fact_3929_frac__gt__0__iff,axiom,
    ! [X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X2 ) )
      = ( ~ ( member_rat @ X2 @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_3930_exp__not__eq__zero,axiom,
    ! [X2: real] :
      ( ( exp_real @ X2 )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_3931_exp__less__cancel,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
     => ( ord_less_real @ X2 @ Y3 ) ) ).

% exp_less_cancel
thf(fact_3932_Ints__0,axiom,
    member_real @ zero_zero_real @ ring_1_Ints_real ).

% Ints_0
thf(fact_3933_Ints__0,axiom,
    member_rat @ zero_zero_rat @ ring_1_Ints_rat ).

% Ints_0
thf(fact_3934_Ints__0,axiom,
    member_int @ zero_zero_int @ ring_1_Ints_int ).

% Ints_0
thf(fact_3935_not__exp__less__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_3936_exp__gt__zero,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_gt_zero
thf(fact_3937_exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X5: real] :
          ( ( exp_real @ X5 )
          = Y3 ) ) ).

% exp_total
thf(fact_3938_exp__ge__zero,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_ge_zero
thf(fact_3939_not__exp__le__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_3940_exp__ge__add__one__self,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).

% exp_ge_add_one_self
thf(fact_3941_exp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).

% exp_gt_one
thf(fact_3942_exp__ge__add__one__self__aux,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_3943_Ints__double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( member_real @ A2 @ ring_1_Ints_real )
     => ( ( ( plus_plus_real @ A2 @ A2 )
          = zero_zero_real )
        = ( A2 = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_3944_Ints__double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( member_rat @ A2 @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A2 @ A2 )
          = zero_zero_rat )
        = ( A2 = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_3945_Ints__double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( member_int @ A2 @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A2 @ A2 )
          = zero_zero_int )
        = ( A2 = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_3946_lemma__exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y3 )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
          & ( ( exp_real @ X5 )
            = Y3 ) ) ) ).

% lemma_exp_total
thf(fact_3947_Ints__odd__nonzero,axiom,
    ! [A2: complex] :
      ( ( member_complex @ A2 @ ring_1_Ints_complex )
     => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A2 ) @ A2 )
       != zero_zero_complex ) ) ).

% Ints_odd_nonzero
thf(fact_3948_Ints__odd__nonzero,axiom,
    ! [A2: real] :
      ( ( member_real @ A2 @ ring_1_Ints_real )
     => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 )
       != zero_zero_real ) ) ).

% Ints_odd_nonzero
thf(fact_3949_Ints__odd__nonzero,axiom,
    ! [A2: rat] :
      ( ( member_rat @ A2 @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_3950_Ints__odd__nonzero,axiom,
    ! [A2: int] :
      ( ( member_int @ A2 @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_3951_exp__divide__power__eq,axiom,
    ! [N: nat,X2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
        = ( exp_complex @ X2 ) ) ) ).

% exp_divide_power_eq
thf(fact_3952_exp__divide__power__eq,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
        = ( exp_real @ X2 ) ) ) ).

% exp_divide_power_eq
thf(fact_3953_Ints__odd__less__0,axiom,
    ! [A2: real] :
      ( ( member_real @ A2 @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 ) @ zero_zero_real )
        = ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).

% Ints_odd_less_0
thf(fact_3954_Ints__odd__less__0,axiom,
    ! [A2: rat] :
      ( ( member_rat @ A2 @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 ) @ zero_zero_rat )
        = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_3955_Ints__odd__less__0,axiom,
    ! [A2: int] :
      ( ( member_int @ A2 @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 ) @ zero_zero_int )
        = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_3956_Ints__nonzero__abs__ge1,axiom,
    ! [X2: code_integer] :
      ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
     => ( ( X2 != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_3957_Ints__nonzero__abs__ge1,axiom,
    ! [X2: real] :
      ( ( member_real @ X2 @ ring_1_Ints_real )
     => ( ( X2 != zero_zero_real )
       => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_3958_Ints__nonzero__abs__ge1,axiom,
    ! [X2: rat] :
      ( ( member_rat @ X2 @ ring_1_Ints_rat )
     => ( ( X2 != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_3959_Ints__nonzero__abs__ge1,axiom,
    ! [X2: int] :
      ( ( member_int @ X2 @ ring_1_Ints_int )
     => ( ( X2 != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_3960_Ints__nonzero__abs__less1,axiom,
    ! [X2: code_integer] :
      ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer )
       => ( X2 = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_3961_Ints__nonzero__abs__less1,axiom,
    ! [X2: real] :
      ( ( member_real @ X2 @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
       => ( X2 = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_3962_Ints__nonzero__abs__less1,axiom,
    ! [X2: rat] :
      ( ( member_rat @ X2 @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat )
       => ( X2 = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_3963_Ints__nonzero__abs__less1,axiom,
    ! [X2: int] :
      ( ( member_int @ X2 @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int )
       => ( X2 = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_3964_Ints__eq__abs__less1,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
       => ( ( X2 = Y3 )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_3965_Ints__eq__abs__less1,axiom,
    ! [X2: real,Y3: real] :
      ( ( member_real @ X2 @ ring_1_Ints_real )
     => ( ( member_real @ Y3 @ ring_1_Ints_real )
       => ( ( X2 = Y3 )
          = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ one_one_real ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_3966_Ints__eq__abs__less1,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( member_rat @ X2 @ ring_1_Ints_rat )
     => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
       => ( ( X2 = Y3 )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_3967_Ints__eq__abs__less1,axiom,
    ! [X2: int,Y3: int] :
      ( ( member_int @ X2 @ ring_1_Ints_int )
     => ( ( member_int @ Y3 @ ring_1_Ints_int )
       => ( ( X2 = Y3 )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Y3 ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_3968_frac__neg,axiom,
    ! [X2: real] :
      ( ( ( member_real @ X2 @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
          = zero_zero_real ) )
      & ( ~ ( member_real @ X2 @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
          = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X2 ) ) ) ) ) ).

% frac_neg
thf(fact_3969_frac__neg,axiom,
    ! [X2: rat] :
      ( ( ( member_rat @ X2 @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X2 ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X2 @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X2 ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X2 ) ) ) ) ) ).

% frac_neg
thf(fact_3970_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X2 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_3971_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3972_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_3973_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_3974_artanh__minus__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).

% artanh_minus_real
thf(fact_3975_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_3976_sinh__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( sinh_real @ X2 )
        = zero_zero_real )
      = ( member_real @ ( exp_real @ X2 ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).

% sinh_zero_iff
thf(fact_3977_sinh__zero__iff,axiom,
    ! [X2: complex] :
      ( ( ( sinh_complex @ X2 )
        = zero_zero_complex )
      = ( member_complex @ ( exp_complex @ X2 ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).

% sinh_zero_iff
thf(fact_3978_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_3979_log__of__power__le,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_3980_sinh__real__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_real @ X2 @ Y3 ) ) ).

% sinh_real_less_iff
thf(fact_3981_sinh__real__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% sinh_real_le_iff
thf(fact_3982_ln__less__cancel__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_3983_ln__inj__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ( ln_ln_real @ X2 )
            = ( ln_ln_real @ Y3 ) )
          = ( X2 = Y3 ) ) ) ) ).

% ln_inj_iff
thf(fact_3984_sinh__real__pos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% sinh_real_pos_iff
thf(fact_3985_sinh__real__neg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sinh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_3986_sinh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_3987_sinh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% sinh_real_nonneg_iff
thf(fact_3988_sinh__0,axiom,
    ( ( sinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% sinh_0
thf(fact_3989_ln__le__cancel__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_3990_ln__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_3991_ln__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
        = ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% ln_gt_zero_iff
thf(fact_3992_ln__eq__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ln_ln_real @ X2 )
          = zero_zero_real )
        = ( X2 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_3993_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_3994_Gcd__empty,axiom,
    ( ( gcd_Gcd_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_empty
thf(fact_3995_exp__ln,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( exp_real @ ( ln_ln_real @ X2 ) )
        = X2 ) ) ).

% exp_ln
thf(fact_3996_exp__ln__iff,axiom,
    ! [X2: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X2 ) )
        = X2 )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% exp_ln_iff
thf(fact_3997_ln__ge__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
        = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% ln_ge_zero_iff
thf(fact_3998_ln__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_3999_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
          = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_4000_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
          = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_4001_one__less__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A2 @ X2 ) )
          = ( ord_less_real @ A2 @ X2 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_4002_log__less__one__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ one_one_real )
          = ( ord_less_real @ X2 @ A2 ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_4003_log__less__cancel__iff,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) )
            = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_4004_log__eq__one,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ A2 @ A2 )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_4005_Gcd__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( gcd_Gcd_nat @ A3 )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_4006_Gcd__0__iff,axiom,
    ! [A3: set_int] :
      ( ( ( gcd_Gcd_int @ A3 )
        = zero_zero_int )
      = ( ord_less_eq_set_int @ A3 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).

% Gcd_0_iff
thf(fact_4007_log__le__cancel__iff,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) )
            = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_4008_log__le__one__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ one_one_real )
          = ( ord_less_eq_real @ X2 @ A2 ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_4009_one__le__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A2 @ X2 ) )
          = ( ord_less_eq_real @ A2 @ X2 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_4010_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_4011_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
          = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_4012_log__pow__cancel,axiom,
    ! [A2: real,B3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ A2 @ ( power_power_real @ A2 @ B3 ) )
          = ( semiri5074537144036343181t_real @ B3 ) ) ) ) ).

% log_pow_cancel
thf(fact_4013_ln__less__self,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).

% ln_less_self
thf(fact_4014_ln__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).

% ln_bound
thf(fact_4015_ln__gt__zero__imp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_4016_ln__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_4017_ln__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).

% ln_gt_zero
thf(fact_4018_ln__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).

% ln_ge_zero
thf(fact_4019_ln__ge__zero__imp__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_4020_ln__add__one__self__le__self,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self
thf(fact_4021_ln__eq__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ln_ln_real @ X2 )
          = ( minus_minus_real @ X2 @ one_one_real ) )
       => ( X2 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_4022_ln__div,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y3 ) )
          = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_div
thf(fact_4023_log__base__change,axiom,
    ! [A2: real,B3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ B3 @ X2 )
          = ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ B3 ) ) ) ) ) ).

% log_base_change
thf(fact_4024_ln__ge__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X2 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X2 ) ) ) ).

% ln_ge_iff
thf(fact_4025_ln__x__over__x__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_4026_less__log__of__power,axiom,
    ! [B3: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B3 @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).

% less_log_of_power
thf(fact_4027_log__of__power__eq,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B3 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_4028_ln__le__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_4029_ln__diff__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y3 ) @ Y3 ) ) ) ) ).

% ln_diff_le
thf(fact_4030_ln__add__one__self__le__self2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self2
thf(fact_4031_log__divide,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A2 @ ( divide_divide_real @ X2 @ Y3 ) )
              = ( minus_minus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_4032_le__log__of__power,axiom,
    ! [B3: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B3 @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).

% le_log_of_power
thf(fact_4033_log__base__pow,axiom,
    ! [A2: real,N: nat,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( log @ ( power_power_real @ A2 @ N ) @ X2 )
        = ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_4034_ln__one__minus__pos__upper__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_4035_log__of__power__less,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_4036_log__root,axiom,
    ! [N: nat,A2: real,B3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ( log @ B3 @ ( root @ N @ A2 ) )
          = ( divide_divide_real @ ( log @ B3 @ A2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_4037_decr__lemma,axiom,
    ! [D: int,X2: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_4038_incr__lemma,axiom,
    ! [D: int,Z: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_4039_Bernoulli__inequality,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_4040_linear__plus__1__le__power,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_4041_find__Some__iff,axiom,
    ! [P: int > $o,Xs: list_int,X2: int] :
      ( ( ( find_int @ P @ Xs )
        = ( some_int @ X2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I4 ) )
            & ( X2
              = ( nth_int @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_4042_find__Some__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = ( some_P7363390416028606310at_nat @ X2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ( X2
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_4043_find__Some__iff,axiom,
    ! [P: num > $o,Xs: list_num,X2: num] :
      ( ( ( find_num @ P @ Xs )
        = ( some_num @ X2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I4 ) )
            & ( X2
              = ( nth_num @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_4044_find__Some__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = ( some_VEBT_VEBT @ X2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ( X2
              = ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_4045_find__Some__iff,axiom,
    ! [P: $o > $o,Xs: list_o,X2: $o] :
      ( ( ( find_o @ P @ Xs )
        = ( some_o @ X2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
            & ( P @ ( nth_o @ Xs @ I4 ) )
            & ( X2
              = ( nth_o @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_4046_find__Some__iff,axiom,
    ! [P: nat > $o,Xs: list_nat,X2: nat] :
      ( ( ( find_nat @ P @ Xs )
        = ( some_nat @ X2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I4 ) )
            & ( X2
              = ( nth_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_4047_find__Some__iff2,axiom,
    ! [X2: int,P: int > $o,Xs: list_int] :
      ( ( ( some_int @ X2 )
        = ( find_int @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I4 ) )
            & ( X2
              = ( nth_int @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_4048_find__Some__iff2,axiom,
    ! [X2: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X2 )
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ( X2
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_4049_find__Some__iff2,axiom,
    ! [X2: num,P: num > $o,Xs: list_num] :
      ( ( ( some_num @ X2 )
        = ( find_num @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I4 ) )
            & ( X2
              = ( nth_num @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_4050_find__Some__iff2,axiom,
    ! [X2: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( some_VEBT_VEBT @ X2 )
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ( X2
              = ( nth_VEBT_VEBT @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_4051_find__Some__iff2,axiom,
    ! [X2: $o,P: $o > $o,Xs: list_o] :
      ( ( ( some_o @ X2 )
        = ( find_o @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
            & ( P @ ( nth_o @ Xs @ I4 ) )
            & ( X2
              = ( nth_o @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_4052_find__Some__iff2,axiom,
    ! [X2: nat,P: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X2 )
        = ( find_nat @ P @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I4 ) )
            & ( X2
              = ( nth_nat @ Xs @ I4 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I4 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_4053_mult__cancel__right,axiom,
    ! [A2: complex,C: complex,B3: complex] :
      ( ( ( times_times_complex @ A2 @ C )
        = ( times_times_complex @ B3 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_4054_mult__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ( times_times_real @ A2 @ C )
        = ( times_times_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_4055_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ( times_times_rat @ A2 @ C )
        = ( times_times_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_4056_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ( times_times_nat @ A2 @ C )
        = ( times_times_nat @ B3 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_4057_mult__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ( times_times_int @ A2 @ C )
        = ( times_times_int @ B3 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_4058_mult__cancel__left,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( ( times_times_complex @ C @ A2 )
        = ( times_times_complex @ C @ B3 ) )
      = ( ( C = zero_zero_complex )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_4059_mult__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ( times_times_real @ C @ A2 )
        = ( times_times_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_4060_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ( times_times_rat @ C @ A2 )
        = ( times_times_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_4061_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ( times_times_nat @ C @ A2 )
        = ( times_times_nat @ C @ B3 ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_4062_mult__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ( times_times_int @ C @ A2 )
        = ( times_times_int @ C @ B3 ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_4063_mult__eq__0__iff,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( times_times_complex @ A2 @ B3 )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        | ( B3 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_4064_mult__eq__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( times_times_real @ A2 @ B3 )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_4065_mult__eq__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( times_times_rat @ A2 @ B3 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_4066_mult__eq__0__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( times_times_nat @ A2 @ B3 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        | ( B3 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_4067_mult__eq__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ( times_times_int @ A2 @ B3 )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        | ( B3 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_4068_mult__zero__right,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ A2 @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_4069_mult__zero__right,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_4070_mult__zero__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_4071_mult__zero__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_4072_mult__zero__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_4073_mult__zero__left,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A2 )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_4074_mult__zero__left,axiom,
    ! [A2: real] :
      ( ( times_times_real @ zero_zero_real @ A2 )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_4075_mult__zero__left,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_4076_mult__zero__left,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_4077_mult__zero__left,axiom,
    ! [A2: int] :
      ( ( times_times_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_4078_mult_Oright__neutral,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ A2 @ one_one_complex )
      = A2 ) ).

% mult.right_neutral
thf(fact_4079_mult_Oright__neutral,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ one_one_real )
      = A2 ) ).

% mult.right_neutral
thf(fact_4080_mult_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.right_neutral
thf(fact_4081_mult_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.right_neutral
thf(fact_4082_mult_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.right_neutral
thf(fact_4083_mult__1,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ one_one_complex @ A2 )
      = A2 ) ).

% mult_1
thf(fact_4084_mult__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ one_one_real @ A2 )
      = A2 ) ).

% mult_1
thf(fact_4085_mult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_4086_mult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_4087_mult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% mult_1
thf(fact_4088_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_4089_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_4090_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_4091_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_4092_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_4093_sum__squares__eq__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_4094_sum__squares__eq__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
        = zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_4095_sum__squares__eq__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_4096_mult__cancel__right2,axiom,
    ! [A2: complex,C: complex] :
      ( ( ( times_times_complex @ A2 @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_4097_mult__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( ( times_times_real @ A2 @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_4098_mult__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ( times_times_rat @ A2 @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_4099_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ( times_times_int @ A2 @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_4100_mult__cancel__right1,axiom,
    ! [C: complex,B3: complex] :
      ( ( C
        = ( times_times_complex @ B3 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B3 = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_4101_mult__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( C
        = ( times_times_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( B3 = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_4102_mult__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( C
        = ( times_times_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B3 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_4103_mult__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( C
        = ( times_times_int @ B3 @ C ) )
      = ( ( C = zero_zero_int )
        | ( B3 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_4104_mult__cancel__left2,axiom,
    ! [C: complex,A2: complex] :
      ( ( ( times_times_complex @ C @ A2 )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_4105_mult__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( ( times_times_real @ C @ A2 )
        = C )
      = ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_4106_mult__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ( times_times_rat @ C @ A2 )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_4107_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ( times_times_int @ C @ A2 )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_4108_mult__cancel__left1,axiom,
    ! [C: complex,B3: complex] :
      ( ( C
        = ( times_times_complex @ C @ B3 ) )
      = ( ( C = zero_zero_complex )
        | ( B3 = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_4109_mult__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( C
        = ( times_times_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( B3 = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_4110_mult__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( C
        = ( times_times_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( B3 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_4111_mult__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( C
        = ( times_times_int @ C @ B3 ) )
      = ( ( C = zero_zero_int )
        | ( B3 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_4112_div__mult__mult1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
        = ( divide_divide_int @ A2 @ B3 ) ) ) ).

% div_mult_mult1
thf(fact_4113_div__mult__mult1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
        = ( divide_divide_nat @ A2 @ B3 ) ) ) ).

% div_mult_mult1
thf(fact_4114_div__mult__mult2,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ A2 @ B3 ) ) ) ).

% div_mult_mult2
thf(fact_4115_div__mult__mult2,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
        = ( divide_divide_nat @ A2 @ B3 ) ) ) ).

% div_mult_mult2
thf(fact_4116_div__mult__mult1__if,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
          = ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_4117_div__mult__mult1__if,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
          = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_4118_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B3 ) @ A2 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_4119_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ B3 ) @ A2 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_4120_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ A2 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_4121_nonzero__mult__div__cancel__left,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ B3 ) @ A2 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_4122_nonzero__mult__div__cancel__left,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B3 ) @ A2 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_4123_nonzero__mult__div__cancel__right,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_4124_nonzero__mult__div__cancel__right,axiom,
    ! [B3: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_4125_nonzero__mult__div__cancel__right,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_4126_nonzero__mult__div__cancel__right,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_4127_nonzero__mult__div__cancel__right,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B3 ) @ B3 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_4128_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
          = ( divide_divide_rat @ A2 @ B3 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_4129_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
          = ( divide_divide_real @ A2 @ B3 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_4130_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
          = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_4131_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
        = ( divide_divide_rat @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_4132_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
        = ( divide_divide_real @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_4133_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
        = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_4134_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ B3 @ C ) )
        = ( divide_divide_rat @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_4135_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B3 @ C ) )
        = ( divide_divide_real @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_4136_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ B3 @ C ) )
        = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_4137_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
        = ( divide_divide_rat @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_4138_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
        = ( divide_divide_real @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_4139_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) )
        = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_4140_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ C @ B3 ) )
        = ( divide_divide_rat @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_4141_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B3 ) )
        = ( divide_divide_real @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_4142_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ C @ B3 ) )
        = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_4143_not__real__square__gt__zero,axiom,
    ! [X2: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
      = ( X2 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_4144_real__root__Suc__0,axiom,
    ! [X2: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
      = X2 ) ).

% real_root_Suc_0
thf(fact_4145_real__root__eq__iff,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X2 )
          = ( root @ N @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% real_root_eq_iff
thf(fact_4146_root__0,axiom,
    ! [X2: real] :
      ( ( root @ zero_zero_nat @ X2 )
      = zero_zero_real ) ).

% root_0
thf(fact_4147_div__mult__self1,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B3 ) ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% div_mult_self1
thf(fact_4148_div__mult__self1,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% div_mult_self1
thf(fact_4149_div__mult__self2,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B3 @ C ) ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% div_mult_self2
thf(fact_4150_div__mult__self2,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% div_mult_self2
thf(fact_4151_div__mult__self3,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A2 ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% div_mult_self3
thf(fact_4152_div__mult__self3,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A2 ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% div_mult_self3
thf(fact_4153_div__mult__self4,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A2 ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% div_mult_self4
thf(fact_4154_div__mult__self4,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A2 ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% div_mult_self4
thf(fact_4155_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( divide_divide_rat @ B3 @ ( times_times_rat @ A2 @ B3 ) )
        = ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_4156_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ B3 @ ( times_times_real @ A2 @ B3 ) )
        = ( divide_divide_real @ one_one_real @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_4157_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B3 @ ( times_times_complex @ A2 @ B3 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_4158_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B3 ) )
        = ( divide_divide_rat @ one_one_rat @ B3 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_4159_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B3 ) )
        = ( divide_divide_real @ one_one_real @ B3 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_4160_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A2 @ ( times_times_complex @ A2 @ B3 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B3 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_4161_real__root__eq__0__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X2 )
          = zero_zero_real )
        = ( X2 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_4162_real__root__less__iff,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ X2 @ Y3 ) ) ) ).

% real_root_less_iff
thf(fact_4163_real__root__le__iff,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).

% real_root_le_iff
thf(fact_4164_real__root__eq__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X2 )
          = one_one_real )
        = ( X2 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_4165_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_4166_real__root__lt__0__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X2 ) @ zero_zero_real )
        = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_4167_real__root__gt__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_gt_0_iff
thf(fact_4168_real__root__le__0__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_4169_real__root__ge__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_ge_0_iff
thf(fact_4170_real__root__lt__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_4171_real__root__gt__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).

% real_root_gt_1_iff
thf(fact_4172_real__root__le__1__iff,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_4173_real__root__ge__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).

% real_root_ge_1_iff
thf(fact_4174_real__root__pow__pos2,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
          = X2 ) ) ) ).

% real_root_pow_pos2
thf(fact_4175_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: complex,B3: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
      = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4176_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A2 @ B3 ) @ C )
      = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4177_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B3 ) @ C )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4178_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4179_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B3 ) @ C )
      = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_4180_mult_Oassoc,axiom,
    ! [A2: complex,B3: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
      = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).

% mult.assoc
thf(fact_4181_mult_Oassoc,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A2 @ B3 ) @ C )
      = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).

% mult.assoc
thf(fact_4182_mult_Oassoc,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B3 ) @ C )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).

% mult.assoc
thf(fact_4183_mult_Oassoc,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).

% mult.assoc
thf(fact_4184_mult_Oassoc,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B3 ) @ C )
      = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).

% mult.assoc
thf(fact_4185_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_4186_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_4187_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_4188_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_4189_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).

% mult.commute
thf(fact_4190_mult_Oleft__commute,axiom,
    ! [B3: complex,A2: complex,C: complex] :
      ( ( times_times_complex @ B3 @ ( times_times_complex @ A2 @ C ) )
      = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).

% mult.left_commute
thf(fact_4191_mult_Oleft__commute,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( times_times_real @ B3 @ ( times_times_real @ A2 @ C ) )
      = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).

% mult.left_commute
thf(fact_4192_mult_Oleft__commute,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( times_times_rat @ B3 @ ( times_times_rat @ A2 @ C ) )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).

% mult.left_commute
thf(fact_4193_mult_Oleft__commute,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( times_times_nat @ B3 @ ( times_times_nat @ A2 @ C ) )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).

% mult.left_commute
thf(fact_4194_mult_Oleft__commute,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( times_times_int @ B3 @ ( times_times_int @ A2 @ C ) )
      = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).

% mult.left_commute
thf(fact_4195_mult__right__cancel,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A2 @ C )
          = ( times_times_complex @ B3 @ C ) )
        = ( A2 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_4196_mult__right__cancel,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A2 @ C )
          = ( times_times_real @ B3 @ C ) )
        = ( A2 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_4197_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C )
          = ( times_times_rat @ B3 @ C ) )
        = ( A2 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_4198_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A2 @ C )
          = ( times_times_nat @ B3 @ C ) )
        = ( A2 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_4199_mult__right__cancel,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A2 @ C )
          = ( times_times_int @ B3 @ C ) )
        = ( A2 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_4200_mult__left__cancel,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A2 )
          = ( times_times_complex @ C @ B3 ) )
        = ( A2 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_4201_mult__left__cancel,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A2 )
          = ( times_times_real @ C @ B3 ) )
        = ( A2 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_4202_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A2 )
          = ( times_times_rat @ C @ B3 ) )
        = ( A2 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_4203_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A2 )
          = ( times_times_nat @ C @ B3 ) )
        = ( A2 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_4204_mult__left__cancel,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A2 )
          = ( times_times_int @ C @ B3 ) )
        = ( A2 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_4205_no__zero__divisors,axiom,
    ! [A2: complex,B3: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( B3 != zero_zero_complex )
       => ( ( times_times_complex @ A2 @ B3 )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_4206_no__zero__divisors,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 != zero_zero_real )
     => ( ( B3 != zero_zero_real )
       => ( ( times_times_real @ A2 @ B3 )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_4207_no__zero__divisors,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B3 != zero_zero_rat )
       => ( ( times_times_rat @ A2 @ B3 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_4208_no__zero__divisors,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B3 != zero_zero_nat )
       => ( ( times_times_nat @ A2 @ B3 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_4209_no__zero__divisors,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B3 != zero_zero_int )
       => ( ( times_times_int @ A2 @ B3 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_4210_divisors__zero,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( times_times_complex @ A2 @ B3 )
        = zero_zero_complex )
     => ( ( A2 = zero_zero_complex )
        | ( B3 = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_4211_divisors__zero,axiom,
    ! [A2: real,B3: real] :
      ( ( ( times_times_real @ A2 @ B3 )
        = zero_zero_real )
     => ( ( A2 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_4212_divisors__zero,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( times_times_rat @ A2 @ B3 )
        = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_4213_divisors__zero,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( times_times_nat @ A2 @ B3 )
        = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B3 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_4214_divisors__zero,axiom,
    ! [A2: int,B3: int] :
      ( ( ( times_times_int @ A2 @ B3 )
        = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B3 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_4215_mult__not__zero,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( times_times_complex @ A2 @ B3 )
       != zero_zero_complex )
     => ( ( A2 != zero_zero_complex )
        & ( B3 != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_4216_mult__not__zero,axiom,
    ! [A2: real,B3: real] :
      ( ( ( times_times_real @ A2 @ B3 )
       != zero_zero_real )
     => ( ( A2 != zero_zero_real )
        & ( B3 != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_4217_mult__not__zero,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( times_times_rat @ A2 @ B3 )
       != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B3 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_4218_mult__not__zero,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( times_times_nat @ A2 @ B3 )
       != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B3 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_4219_mult__not__zero,axiom,
    ! [A2: int,B3: int] :
      ( ( ( times_times_int @ A2 @ B3 )
       != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B3 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_4220_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ one_one_complex @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_4221_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ one_one_real @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_4222_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_4223_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_4224_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_4225_mult_Ocomm__neutral,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ A2 @ one_one_complex )
      = A2 ) ).

% mult.comm_neutral
thf(fact_4226_mult_Ocomm__neutral,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ one_one_real )
      = A2 ) ).

% mult.comm_neutral
thf(fact_4227_mult_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_4228_mult_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_4229_mult_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.comm_neutral
thf(fact_4230_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X2 ) @ Y3 )
      = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_4231_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_4232_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y3 )
      = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_4233_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y3 )
      = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_4234_mult__of__nat__commute,axiom,
    ! [X2: nat,Y3: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y3 )
      = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_4235_real__root__pos__pos__le,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ).

% real_root_pos_pos_le
thf(fact_4236_mult__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_4237_mult__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_4238_mult__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_4239_mult__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_4240_mult__mono_H,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4241_mult__mono_H,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4242_mult__mono_H,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4243_mult__mono_H,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_4244_zero__le__square,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_4245_zero__le__square,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_4246_zero__le__square,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_4247_split__mult__pos__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ).

% split_mult_pos_le
thf(fact_4248_split__mult__pos__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ).

% split_mult_pos_le
thf(fact_4249_split__mult__pos__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ).

% split_mult_pos_le
thf(fact_4250_mult__left__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_4251_mult__left__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_4252_mult__left__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_4253_mult__nonpos__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_4254_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_4255_mult__nonpos__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_4256_mult__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_4257_mult__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_4258_mult__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_4259_mult__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_4260_mult__right__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_4261_mult__right__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_4262_mult__right__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_4263_mult__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4264_mult__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4265_mult__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4266_mult__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_4267_mult__le__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).

% mult_le_0_iff
thf(fact_4268_mult__le__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% mult_le_0_iff
thf(fact_4269_mult__le__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ) ) ).

% mult_le_0_iff
thf(fact_4270_split__mult__neg__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_4271_split__mult__neg__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_4272_split__mult__neg__le,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
          & ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_4273_split__mult__neg__le,axiom,
    ! [A2: int,B3: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_4274_mult__nonneg__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4275_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4276_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4277_mult__nonneg__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_4278_mult__nonneg__nonpos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4279_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4280_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4281_mult__nonneg__nonpos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_4282_mult__nonpos__nonneg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4283_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4284_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4285_mult__nonpos__nonneg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_4286_mult__nonneg__nonpos2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B3 @ A2 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4287_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B3 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4288_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4289_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B3 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_4290_zero__le__mult__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_4291_zero__le__mult__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_4292_zero__le__mult__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_4293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4295_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4296_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_4297_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4298_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4299_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4300_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_4301_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A2 @ B3 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_4302_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A2 @ B3 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_4303_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B3 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_4304_mult__strict__right__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4305_mult__strict__right__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4306_mult__strict__right__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4307_mult__strict__right__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_4308_mult__strict__right__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_4309_mult__strict__right__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_4310_mult__strict__right__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_4311_mult__less__cancel__left__disj,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A2 @ B3 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_4312_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A2 @ B3 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_4313_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B3 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_4314_mult__strict__left__mono,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4315_mult__strict__left__mono,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4316_mult__strict__left__mono,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4317_mult__strict__left__mono,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_4318_mult__strict__left__mono__neg,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_4319_mult__strict__left__mono__neg,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_4320_mult__strict__left__mono__neg,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_4321_mult__less__cancel__left__pos,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_real @ A2 @ B3 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_4322_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_rat @ A2 @ B3 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_4323_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_int @ A2 @ B3 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_4324_mult__less__cancel__left__neg,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_real @ B3 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_4325_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_rat @ B3 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_4326_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_int @ B3 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_4327_zero__less__mult__pos2,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4328_zero__less__mult__pos2,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B3 @ A2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4329_zero__less__mult__pos2,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4330_zero__less__mult__pos2,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_4331_zero__less__mult__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_4332_zero__less__mult__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_4333_zero__less__mult__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_4334_zero__less__mult__pos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_4335_zero__less__mult__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_4336_zero__less__mult__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_4337_zero__less__mult__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ zero_zero_int @ B3 ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_4338_mult__pos__neg2,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B3 @ A2 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_4339_mult__pos__neg2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B3 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_4340_mult__pos__neg2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_4341_mult__pos__neg2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B3 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_4342_mult__pos__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_4343_mult__pos__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_4344_mult__pos__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_4345_mult__pos__pos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_4346_mult__pos__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_4347_mult__pos__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_4348_mult__pos__neg,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_4349_mult__pos__neg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_4350_mult__neg__pos,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_4351_mult__neg__pos,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_4352_mult__neg__pos,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_4353_mult__neg__pos,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_4354_mult__less__0__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).

% mult_less_0_iff
thf(fact_4355_mult__less__0__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% mult_less_0_iff
thf(fact_4356_mult__less__0__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ B3 @ zero_zero_int ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).

% mult_less_0_iff
thf(fact_4357_not__square__less__zero,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_4358_not__square__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_4359_not__square__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_4360_mult__neg__neg,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).

% mult_neg_neg
thf(fact_4361_mult__neg__neg,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).

% mult_neg_neg
thf(fact_4362_mult__neg__neg,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).

% mult_neg_neg
thf(fact_4363_add__scale__eq__noteq,axiom,
    ! [R2: complex,A2: complex,B3: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A2 @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B3 @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4364_add__scale__eq__noteq,axiom,
    ! [R2: real,A2: real,B3: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( ( plus_plus_real @ A2 @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B3 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4365_add__scale__eq__noteq,axiom,
    ! [R2: rat,A2: rat,B3: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B3 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4366_add__scale__eq__noteq,axiom,
    ! [R2: nat,A2: nat,B3: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B3 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4367_add__scale__eq__noteq,axiom,
    ! [R2: int,A2: int,B3: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A2 = B3 )
          & ( C != D ) )
       => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B3 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4368_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_4369_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_4370_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_4371_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_4372_frac__eq__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X2 @ Y3 )
            = ( divide_divide_rat @ W2 @ Z ) )
          = ( ( times_times_rat @ X2 @ Z )
            = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_4373_frac__eq__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X2 @ Y3 )
            = ( divide_divide_real @ W2 @ Z ) )
          = ( ( times_times_real @ X2 @ Z )
            = ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_4374_frac__eq__eq,axiom,
    ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X2 @ Y3 )
            = ( divide1717551699836669952omplex @ W2 @ Z ) )
          = ( ( times_times_complex @ X2 @ Z )
            = ( times_times_complex @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_4375_divide__eq__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B3 @ C )
        = A2 )
      = ( ( ( C != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ A2 @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_4376_divide__eq__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ( divide_divide_real @ B3 @ C )
        = A2 )
      = ( ( ( C != zero_zero_real )
         => ( B3
            = ( times_times_real @ A2 @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_4377_divide__eq__eq,axiom,
    ! [B3: complex,C: complex,A2: complex] :
      ( ( ( divide1717551699836669952omplex @ B3 @ C )
        = A2 )
      = ( ( ( C != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ A2 @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_4378_eq__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C )
            = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_4379_eq__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( A2
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A2 @ C )
            = B3 ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_4380_eq__divide__eq,axiom,
    ! [A2: complex,B3: complex,C: complex] :
      ( ( A2
        = ( divide1717551699836669952omplex @ B3 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ C )
            = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_4381_divide__eq__imp,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B3
          = ( times_times_rat @ A2 @ C ) )
       => ( ( divide_divide_rat @ B3 @ C )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_4382_divide__eq__imp,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( B3
          = ( times_times_real @ A2 @ C ) )
       => ( ( divide_divide_real @ B3 @ C )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_4383_divide__eq__imp,axiom,
    ! [C: complex,B3: complex,A2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B3
          = ( times_times_complex @ A2 @ C ) )
       => ( ( divide1717551699836669952omplex @ B3 @ C )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_4384_eq__divide__imp,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C )
          = B3 )
       => ( A2
          = ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_4385_eq__divide__imp,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A2 @ C )
          = B3 )
       => ( A2
          = ( divide_divide_real @ B3 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_4386_eq__divide__imp,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A2 @ C )
          = B3 )
       => ( A2
          = ( divide1717551699836669952omplex @ B3 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_4387_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B3 @ C )
          = A2 )
        = ( B3
          = ( times_times_rat @ A2 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_4388_nonzero__divide__eq__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B3 @ C )
          = A2 )
        = ( B3
          = ( times_times_real @ A2 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_4389_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B3: complex,A2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B3 @ C )
          = A2 )
        = ( B3
          = ( times_times_complex @ A2 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_4390_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A2
          = ( divide_divide_rat @ B3 @ C ) )
        = ( ( times_times_rat @ A2 @ C )
          = B3 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_4391_nonzero__eq__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( A2
          = ( divide_divide_real @ B3 @ C ) )
        = ( ( times_times_real @ A2 @ C )
          = B3 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_4392_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A2: complex,B3: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A2
          = ( divide1717551699836669952omplex @ B3 @ C ) )
        = ( ( times_times_complex @ A2 @ C )
          = B3 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_4393_abs__mult__less,axiom,
    ! [A2: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B3 ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4394_abs__mult__less,axiom,
    ! [A2: real,C: real,B3: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B3 ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4395_abs__mult__less,axiom,
    ! [A2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B3 ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4396_abs__mult__less,axiom,
    ! [A2: int,C: int,B3: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B3 ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_4397_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_4398_real__minus__mult__self__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).

% real_minus_mult_self_le
thf(fact_4399_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_4400_real__root__less__mono,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X2 @ Y3 )
       => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).

% real_root_less_mono
thf(fact_4401_real__root__le__mono,axiom,
    ! [N: nat,X2: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).

% real_root_le_mono
thf(fact_4402_real__root__power,axiom,
    ! [N: nat,X2: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X2 @ K ) )
        = ( power_power_real @ ( root @ N @ X2 ) @ K ) ) ) ).

% real_root_power
thf(fact_4403_real__root__abs,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X2 ) )
        = ( abs_abs_real @ ( root @ N @ X2 ) ) ) ) ).

% real_root_abs
thf(fact_4404_log__base__root,axiom,
    ! [N: nat,B3: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ( log @ ( root @ N @ B3 ) @ X2 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ) ).

% log_base_root
thf(fact_4405_mult__less__le__imp__less,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4406_mult__less__le__imp__less,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4407_mult__less__le__imp__less,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4408_mult__less__le__imp__less,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_4409_mult__le__less__imp__less,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4410_mult__le__less__imp__less,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4411_mult__le__less__imp__less,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4412_mult__le__less__imp__less,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_4413_mult__right__le__imp__le,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4414_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4415_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4416_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_4417_mult__left__le__imp__le,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4418_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4419_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4420_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_4421_mult__le__cancel__left__pos,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_4422_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_4423_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_4424_mult__le__cancel__left__neg,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_4425_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_4426_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_eq_int @ B3 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_4427_mult__less__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_4428_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_4429_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_4430_mult__strict__mono_H,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4431_mult__strict__mono_H,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4432_mult__strict__mono_H,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4433_mult__strict__mono_H,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_4434_mult__right__less__imp__less,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A2 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4435_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A2 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4436_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4437_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_4438_mult__less__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_4439_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_4440_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_4441_mult__strict__mono,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4442_mult__strict__mono,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4443_mult__strict__mono,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4444_mult__strict__mono,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_4445_mult__left__less__imp__less,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A2 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4446_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A2 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4447_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4448_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_4449_mult__le__cancel__right,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_4450_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_4451_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_4452_mult__le__cancel__left,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_4453_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_4454_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_4455_sum__squares__le__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_4456_sum__squares__le__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_4457_sum__squares__le__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_4458_sum__squares__ge__zero,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_4459_sum__squares__ge__zero,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_4460_sum__squares__ge__zero,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_4461_mult__left__le__one__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_4462_mult__left__le__one__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_4463_mult__left__le__one__le,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_4464_mult__right__le__one__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y3 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_4465_mult__right__le__one__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y3 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_4466_mult__right__le__one__le,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y3 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_4467_mult__le__one,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ A2 @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ( ord_less_eq_real @ B3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_4468_mult__le__one,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ( ord_less_eq_rat @ B3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_4469_mult__le__one,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_eq_nat @ B3 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_4470_mult__le__one,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_4471_mult__left__le,axiom,
    ! [C: real,A2: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4472_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4473_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4474_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_4475_sum__squares__gt__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
      = ( ( X2 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_4476_sum__squares__gt__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
      = ( ( X2 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_4477_sum__squares__gt__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
      = ( ( X2 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_4478_not__sum__squares__lt__zero,axiom,
    ! [X2: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_4479_not__sum__squares__lt__zero,axiom,
    ! [X2: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_4480_not__sum__squares__lt__zero,axiom,
    ! [X2: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_4481_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B3 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B3 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4482_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_4483_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_4484_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_4485_divide__strict__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_4486_divide__strict__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_4487_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_4488_mult__imp__less__div__pos,axiom,
    ! [Y3: real,Z: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_4489_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_4490_mult__imp__div__pos__less,axiom,
    ! [Y3: real,X2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_4491_pos__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).

% pos_less_divide_eq
thf(fact_4492_pos__less__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).

% pos_less_divide_eq
thf(fact_4493_pos__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
        = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_4494_pos__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
        = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_4495_neg__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_4496_neg__less__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_4497_neg__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).

% neg_divide_less_eq
thf(fact_4498_neg__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).

% neg_divide_less_eq
thf(fact_4499_less__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_4500_less__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_4501_divide__less__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_4502_divide__less__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_4503_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4504_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4505_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4506_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4507_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4508_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4509_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
          = ( divide_divide_rat @ ( plus_plus_rat @ A2 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_4510_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
          = ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_4511_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A2 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_4512_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_4513_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_4514_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_4515_add__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_4516_add__frac__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_4517_add__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_4518_add__frac__num,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_4519_add__frac__num,axiom,
    ! [Y3: real,X2: real,Z: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_4520_add__frac__num,axiom,
    ! [Y3: complex,X2: complex,Z: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_4521_add__num__frac,axiom,
    ! [Y3: rat,Z: rat,X2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_4522_add__num__frac,axiom,
    ! [Y3: real,Z: real,X2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_4523_add__num__frac,axiom,
    ! [Y3: complex,Z: complex,X2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_4524_add__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_4525_add__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_4526_add__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_4527_divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_4528_divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_4529_divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_4530_less__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_4531_less__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_4532_less__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_4533_less__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_4534_less__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_4535_less__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_4536_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4537_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4538_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4539_diff__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4540_diff__frac__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4541_diff__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4542_diff__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4543_diff__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4544_diff__divide__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4545_divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4546_divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4547_divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4548_ex__less__of__nat__mult,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X2 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4549_ex__less__of__nat__mult,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4550_eq__minus__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( A2
        = ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A2 @ C )
            = ( uminus_uminus_real @ B3 ) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4551_eq__minus__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C )
            = ( uminus_uminus_rat @ B3 ) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4552_eq__minus__divide__eq,axiom,
    ! [A2: complex,B3: complex,C: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ C )
            = ( uminus1482373934393186551omplex @ B3 ) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4553_minus__divide__eq__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) )
        = A2 )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B3 )
            = ( times_times_real @ A2 @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4554_minus__divide__eq__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) )
        = A2 )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B3 )
            = ( times_times_rat @ A2 @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4555_minus__divide__eq__eq,axiom,
    ! [B3: complex,C: complex,A2: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) )
        = A2 )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B3 )
            = ( times_times_complex @ A2 @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4556_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( B3 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) )
          = C )
        = ( ( uminus_uminus_real @ A2 )
          = ( times_times_real @ C @ B3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4557_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) )
          = C )
        = ( ( uminus_uminus_rat @ A2 )
          = ( times_times_rat @ C @ B3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4558_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: complex,A2: complex,C: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A2 )
          = ( times_times_complex @ C @ B3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4559_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) ) )
        = ( ( times_times_real @ C @ B3 )
          = ( uminus_uminus_real @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4560_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) ) )
        = ( ( times_times_rat @ C @ B3 )
          = ( uminus_uminus_rat @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4561_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: complex,C: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
        = ( ( times_times_complex @ C @ B3 )
          = ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4562_power__gt1__lemma,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4563_power__gt1__lemma,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4564_power__gt1__lemma,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4565_power__gt1__lemma,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_4566_power__less__power__Suc,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4567_power__less__power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4568_power__less__power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4569_power__less__power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_4570_abs__mult__pos,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X2 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_4571_abs__mult__pos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X2 )
        = ( abs_abs_real @ ( times_times_real @ Y3 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_4572_abs__mult__pos,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X2 )
        = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_4573_abs__mult__pos,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X2 )
        = ( abs_abs_int @ ( times_times_int @ Y3 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_4574_abs__eq__mult,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          | ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
          | ( ord_le3102999989581377725nteger @ B3 @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B3 ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_4575_abs__eq__mult,axiom,
    ! [A2: real,B3: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          | ( ord_less_eq_real @ A2 @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B3 )
          | ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A2 @ B3 ) )
        = ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_4576_abs__eq__mult,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          | ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
          | ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B3 ) )
        = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_4577_abs__eq__mult,axiom,
    ! [A2: int,B3: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          | ( ord_less_eq_int @ A2 @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B3 )
          | ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A2 @ B3 ) )
        = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_4578_ln__mult,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y3 ) )
          = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_mult
thf(fact_4579_reals__Archimedean3,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ! [Y5: real] :
        ? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).

% reals_Archimedean3
thf(fact_4580_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_4581_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K2: int] :
            ( ( P1 @ X5 )
            = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X5: int] :
              ( ( ord_less_int @ X5 @ Z5 )
             => ( ( P @ X5 )
                = ( P1 @ X5 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_4582_plusinfinity,axiom,
    ! [D: int,P4: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K2: int] :
            ( ( P4 @ X5 )
            = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X5: int] :
              ( ( ord_less_int @ Z5 @ X5 )
             => ( ( P @ X5 )
                = ( P4 @ X5 ) ) )
         => ( ? [X_12: int] : ( P4 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_4583_zdiv__zmult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_4584_find__None__iff2,axiom,
    ! [P: real > $o,Xs: list_real] :
      ( ( none_real
        = ( find_real @ P @ Xs ) )
      = ( ~ ? [X: real] :
              ( ( member_real @ X @ ( set_real2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4585_find__None__iff2,axiom,
    ! [P: $o > $o,Xs: list_o] :
      ( ( none_o
        = ( find_o @ P @ Xs ) )
      = ( ~ ? [X: $o] :
              ( ( member_o @ X @ ( set_o2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4586_find__None__iff2,axiom,
    ! [P: set_nat > $o,Xs: list_set_nat] :
      ( ( none_set_nat
        = ( find_set_nat @ P @ Xs ) )
      = ( ~ ? [X: set_nat] :
              ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4587_find__None__iff2,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( none_int
        = ( find_int @ P @ Xs ) )
      = ( ~ ? [X: int] :
              ( ( member_int @ X @ ( set_int2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4588_find__None__iff2,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ~ ? [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4589_find__None__iff2,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( none_nat
        = ( find_nat @ P @ Xs ) )
      = ( ~ ? [X: nat] :
              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4590_find__None__iff2,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ~ ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4591_find__None__iff2,axiom,
    ! [P: num > $o,Xs: list_num] :
      ( ( none_num
        = ( find_num @ P @ Xs ) )
      = ( ~ ? [X: num] :
              ( ( member_num @ X @ ( set_num2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_4592_find__None__iff,axiom,
    ! [P: real > $o,Xs: list_real] :
      ( ( ( find_real @ P @ Xs )
        = none_real )
      = ( ~ ? [X: real] :
              ( ( member_real @ X @ ( set_real2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4593_find__None__iff,axiom,
    ! [P: $o > $o,Xs: list_o] :
      ( ( ( find_o @ P @ Xs )
        = none_o )
      = ( ~ ? [X: $o] :
              ( ( member_o @ X @ ( set_o2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4594_find__None__iff,axiom,
    ! [P: set_nat > $o,Xs: list_set_nat] :
      ( ( ( find_set_nat @ P @ Xs )
        = none_set_nat )
      = ( ~ ? [X: set_nat] :
              ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4595_find__None__iff,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( ( find_int @ P @ Xs )
        = none_int )
      = ( ~ ? [X: int] :
              ( ( member_int @ X @ ( set_int2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4596_find__None__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = none_VEBT_VEBT )
      = ( ~ ? [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4597_find__None__iff,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( find_nat @ P @ Xs )
        = none_nat )
      = ( ~ ? [X: nat] :
              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4598_find__None__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = none_P5556105721700978146at_nat )
      = ( ~ ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4599_find__None__iff,axiom,
    ! [P: num > $o,Xs: list_num] :
      ( ( ( find_num @ P @ Xs )
        = none_num )
      = ( ~ ? [X: num] :
              ( ( member_num @ X @ ( set_num2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_4600_real__root__gt__zero,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).

% real_root_gt_zero
thf(fact_4601_real__root__strict__decreasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ( ord_less_real @ one_one_real @ X2 )
         => ( ord_less_real @ ( root @ N6 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_4602_root__abs__power,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
        = ( abs_abs_real @ Y3 ) ) ) ).

% root_abs_power
thf(fact_4603_mult__less__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_4604_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_4605_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_4606_mult__less__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_4607_mult__less__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_4608_mult__less__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_4609_mult__less__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_4610_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_4611_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_4612_mult__less__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_4613_mult__less__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_4614_mult__less__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_4615_mult__le__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_4616_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_4617_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_4618_mult__le__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_4619_mult__le__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_4620_mult__le__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_4621_mult__le__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_4622_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_4623_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_4624_mult__le__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_4625_mult__le__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_4626_mult__le__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_4627_field__le__mult__one__interval,axiom,
    ! [X2: real,Y3: real] :
      ( ! [Z4: real] :
          ( ( ord_less_real @ zero_zero_real @ Z4 )
         => ( ( ord_less_real @ Z4 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X2 ) @ Y3 ) ) )
     => ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_4628_field__le__mult__one__interval,axiom,
    ! [X2: rat,Y3: rat] :
      ( ! [Z4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z4 )
         => ( ( ord_less_rat @ Z4 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X2 ) @ Y3 ) ) )
     => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_4629_divide__left__mono__neg,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_4630_divide__left__mono__neg,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_4631_mult__imp__le__div__pos,axiom,
    ! [Y3: real,Z: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_4632_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_4633_mult__imp__div__pos__le,axiom,
    ! [Y3: real,X2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_4634_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_4635_pos__le__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).

% pos_le_divide_eq
thf(fact_4636_pos__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).

% pos_le_divide_eq
thf(fact_4637_pos__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
        = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_4638_pos__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
        = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_4639_neg__le__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_4640_neg__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_4641_neg__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).

% neg_divide_le_eq
thf(fact_4642_neg__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).

% neg_divide_le_eq
thf(fact_4643_divide__left__mono,axiom,
    ! [B3: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_left_mono
thf(fact_4644_divide__left__mono,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_left_mono
thf(fact_4645_le__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_4646_le__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_4647_divide__le__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_4648_divide__le__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_4649_convex__bound__le,axiom,
    ! [X2: real,A2: real,Y3: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X2 @ A2 )
     => ( ( ord_less_eq_real @ Y3 @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_4650_convex__bound__le,axiom,
    ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X2 @ A2 )
     => ( ( ord_less_eq_rat @ Y3 @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_4651_convex__bound__le,axiom,
    ! [X2: int,A2: int,Y3: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X2 @ A2 )
     => ( ( ord_less_eq_int @ Y3 @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_4652_frac__le__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_4653_frac__le__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_4654_frac__less__eq,axiom,
    ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_4655_frac__less__eq,axiom,
    ! [Y3: real,Z: real,X2: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_4656_power__Suc__less,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ A2 @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4657_power__Suc__less,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4658_power__Suc__less,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4659_power__Suc__less,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_4660_pos__minus__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
        = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4661_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4662_pos__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4663_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4664_neg__minus__divide__less__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4665_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4666_neg__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4667_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4668_minus__divide__less__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4669_minus__divide__less__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4670_less__minus__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4671_less__minus__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4672_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4673_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4674_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4675_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4676_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4677_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4678_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
          = ( uminus_uminus_real @ B3 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4679_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
          = ( uminus_uminus_rat @ B3 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4680_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
          = ( uminus1482373934393186551omplex @ B3 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4681_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A2: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
          = ( uminus_uminus_real @ B3 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
          = ( divide_divide_real @ ( minus_minus_real @ A2 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4682_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A2: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
          = ( uminus_uminus_rat @ B3 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
          = ( divide_divide_rat @ ( minus_minus_rat @ A2 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4683_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A2: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
          = ( uminus1482373934393186551omplex @ B3 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A2 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4684_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4685_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4686_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4687_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_4688_ln__realpow,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( power_power_real @ X2 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_realpow
thf(fact_4689_q__pos__lemma,axiom,
    ! [B7: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B7 )
       => ( ( ord_less_int @ zero_zero_int @ B7 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_4690_zdiv__mono2__lemma,axiom,
    ! [B3: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B7 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B7 )
             => ( ( ord_less_eq_int @ B7 @ B3 )
               => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_4691_zdiv__mono2__neg__lemma,axiom,
    ! [B3: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B7 )
             => ( ( ord_less_eq_int @ B7 @ B3 )
               => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_4692_unique__quotient__lemma,axiom,
    ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B3 )
         => ( ( ord_less_int @ R2 @ B3 )
           => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_4693_unique__quotient__lemma__neg,axiom,
    ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B3 @ R2 )
         => ( ( ord_less_int @ B3 @ R4 )
           => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_4694_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( plus_plus_int @ X5 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_4695_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( minus_minus_int @ X5 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4696_real__root__pos__pos,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).

% real_root_pos_pos
thf(fact_4697_real__root__strict__increasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ X2 @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N6 @ X2 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_4698_real__root__decreasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N6 )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ord_less_eq_real @ ( root @ N6 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_4699_real__root__pow__pos,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
          = X2 ) ) ) ).

% real_root_pow_pos
thf(fact_4700_real__root__pos__unique,axiom,
    ! [N: nat,Y3: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ Y3 @ N )
            = X2 )
         => ( ( root @ N @ X2 )
            = Y3 ) ) ) ) ).

% real_root_pos_unique
thf(fact_4701_real__root__power__cancel,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
          = X2 ) ) ) ).

% real_root_power_cancel
thf(fact_4702_convex__bound__lt,axiom,
    ! [X2: real,A2: real,Y3: real,U: real,V: real] :
      ( ( ord_less_real @ X2 @ A2 )
     => ( ( ord_less_real @ Y3 @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_4703_convex__bound__lt,axiom,
    ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X2 @ A2 )
     => ( ( ord_less_rat @ Y3 @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_4704_convex__bound__lt,axiom,
    ! [X2: int,A2: int,Y3: int,U: int,V: int] :
      ( ( ord_less_int @ X2 @ A2 )
     => ( ( ord_less_int @ Y3 @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_4705_le__minus__divide__eq,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4706_le__minus__divide__eq,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4707_minus__divide__le__eq,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4708_minus__divide__le__eq,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4709_neg__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4710_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4711_neg__minus__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4712_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4713_pos__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4714_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4715_pos__minus__divide__le__eq,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4716_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4717_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S2 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4718_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4719_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4720_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4721_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4722_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4723_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4724_power__minus__mult,axiom,
    ! [N: nat,A2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_complex @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_4725_power__minus__mult,axiom,
    ! [N: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_real @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_4726_power__minus__mult,axiom,
    ! [N: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_rat @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_4727_power__minus__mult,axiom,
    ! [N: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_nat @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_4728_power__minus__mult,axiom,
    ! [N: nat,A2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_int @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_4729_real__archimedian__rdiv__eq__0,axiom,
    ! [X2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
         => ( X2 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_4730_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B3 )
         => ( ( B3 != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X2 )
             => ( ( log @ A2 @ X2 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( ln_ln_real @ A2 ) ) @ ( log @ B3 @ X2 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_4731_log__mult,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A2 @ ( times_times_real @ X2 @ Y3 ) )
              = ( plus_plus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_4732_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) ) ) ) ).

% split_zdiv
thf(fact_4733_int__div__neg__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B3 @ R2 )
         => ( ( divide_divide_int @ A2 @ B3 )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_4734_int__div__pos__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B3 )
         => ( ( divide_divide_int @ A2 @ B3 )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_4735_log__nat__power,axiom,
    ! [X2: real,B3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ B3 @ ( power_power_real @ X2 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ).

% log_nat_power
thf(fact_4736_real__root__increasing,axiom,
    ! [N: nat,N6: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N6 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N6 @ X2 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_4737_ln__root,axiom,
    ! [N: nat,B3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ( ln_ln_real @ ( root @ N @ B3 ) )
          = ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_4738_mult__le__cancel__iff2,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y3 ) )
        = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4739_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y3 ) )
        = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4740_mult__le__cancel__iff2,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y3 ) )
        = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4741_mult__le__cancel__iff1,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
        = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4742_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
        = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4743_mult__le__cancel__iff1,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
        = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4744_mult__less__iff1,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
        = ( ord_less_real @ X2 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_4745_mult__less__iff1,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
        = ( ord_less_rat @ X2 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_4746_mult__less__iff1,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
        = ( ord_less_int @ X2 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_4747_arctan__add,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y3 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_4748_root__powr__inverse,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( root @ N @ X2 )
          = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_4749_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_4750_log__minus__eq__powr,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ ( log @ B3 @ X2 ) @ Y3 )
            = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_4751_split__root,axiom,
    ! [P: real > $o,N: nat,X2: real] :
      ( ( P @ ( root @ N @ X2 ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
                = X2 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_4752_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_4753_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_4754_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_4755_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_4756_sgn__0,axiom,
    ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% sgn_0
thf(fact_4757_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_4758_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_4759_sgn__0,axiom,
    ( ( sgn_sgn_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% sgn_0
thf(fact_4760_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_4761_powr__0,axiom,
    ! [Z: real] :
      ( ( powr_real @ zero_zero_real @ Z )
      = zero_zero_real ) ).

% powr_0
thf(fact_4762_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( ( powr_real @ W2 @ Z )
        = zero_zero_real )
      = ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_4763_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_4764_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_4765_sgn__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_4766_sgn__less,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( sgn_sgn_real @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% sgn_less
thf(fact_4767_sgn__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( sgn_sgn_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% sgn_less
thf(fact_4768_sgn__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( sgn_sgn_int @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% sgn_less
thf(fact_4769_sgn__greater,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A2 ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% sgn_greater
thf(fact_4770_sgn__greater,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% sgn_greater
thf(fact_4771_sgn__greater,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% sgn_greater
thf(fact_4772_sgn__greater,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% sgn_greater
thf(fact_4773_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_4774_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_4775_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_4776_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_4777_powr__zero__eq__one,axiom,
    ! [X2: real] :
      ( ( ( X2 = zero_zero_real )
       => ( ( powr_real @ X2 @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X2 != zero_zero_real )
       => ( ( powr_real @ X2 @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_4778_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_4779_powr__gt__zero,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X2 @ A2 ) )
      = ( X2 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_4780_powr__nonneg__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_4781_powr__less__cancel__iff,axiom,
    ! [X2: real,A2: real,B3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
        = ( ord_less_real @ A2 @ B3 ) ) ) ).

% powr_less_cancel_iff
thf(fact_4782_zero__less__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% zero_less_arctan_iff
thf(fact_4783_arctan__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( arctan @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_4784_zero__le__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_arctan_iff
thf(fact_4785_arctan__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_4786_sgn__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( sgn_sgn_Code_integer @ A2 )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_4787_sgn__pos,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( sgn_sgn_real @ A2 )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_4788_sgn__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( sgn_sgn_rat @ A2 )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_4789_sgn__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( sgn_sgn_int @ A2 )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_4790_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_4791_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_4792_abs__sgn__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
        = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
thf(fact_4793_abs__sgn__eq__1,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
        = one_one_real ) ) ).

% abs_sgn_eq_1
thf(fact_4794_abs__sgn__eq__1,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
        = one_one_rat ) ) ).

% abs_sgn_eq_1
thf(fact_4795_abs__sgn__eq__1,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_4796_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_4797_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_4798_powr__eq__one__iff,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ( powr_real @ A2 @ X2 )
          = one_one_real )
        = ( X2 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_4799_powr__one__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( powr_real @ X2 @ one_one_real )
        = X2 )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% powr_one_gt_zero_iff
thf(fact_4800_powr__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ one_one_real )
        = X2 ) ) ).

% powr_one
thf(fact_4801_powr__le__cancel__iff,axiom,
    ! [X2: real,A2: real,B3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
        = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% powr_le_cancel_iff
thf(fact_4802_sgn__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( sgn_sgn_int @ A2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_4803_sgn__neg,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( sgn_sgn_real @ A2 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% sgn_neg
thf(fact_4804_sgn__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_4805_sgn__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_4806_powr__log__cancel,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( powr_real @ A2 @ ( log @ A2 @ X2 ) )
            = X2 ) ) ) ) ).

% powr_log_cancel
thf(fact_4807_log__powr__cancel,axiom,
    ! [A2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ A2 @ ( powr_real @ A2 @ Y3 ) )
          = Y3 ) ) ) ).

% log_powr_cancel
thf(fact_4808_sgn__0__0,axiom,
    ! [A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
thf(fact_4809_sgn__0__0,axiom,
    ! [A2: real] :
      ( ( ( sgn_sgn_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_4810_sgn__0__0,axiom,
    ! [A2: rat] :
      ( ( ( sgn_sgn_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% sgn_0_0
thf(fact_4811_sgn__0__0,axiom,
    ! [A2: int] :
      ( ( ( sgn_sgn_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_4812_sgn__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
thf(fact_4813_sgn__eq__0__iff,axiom,
    ! [A2: complex] :
      ( ( ( sgn_sgn_complex @ A2 )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_4814_sgn__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( sgn_sgn_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_4815_sgn__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( sgn_sgn_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% sgn_eq_0_iff
thf(fact_4816_sgn__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( sgn_sgn_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_4817_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_4818_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_4819_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_4820_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_4821_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_4822_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_4823_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_4824_arctan__monotone,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone
thf(fact_4825_arctan__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
      = ( ord_less_real @ X2 @ Y3 ) ) ).

% arctan_less_iff
thf(fact_4826_arctan__monotone_H,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ Y3 )
     => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone'
thf(fact_4827_arctan__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
      = ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% arctan_le_iff
thf(fact_4828_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_4829_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_4830_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_4831_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_4832_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_4833_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_4834_powr__non__neg,axiom,
    ! [A2: real,X2: real] :
      ~ ( ord_less_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_4835_powr__less__mono2__neg,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y3 )
         => ( ord_less_real @ ( powr_real @ Y3 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_4836_powr__ge__pzero,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y3 ) ) ).

% powr_ge_pzero
thf(fact_4837_powr__mono2,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).

% powr_mono2
thf(fact_4838_powr__less__mono,axiom,
    ! [A2: real,B3: real,X2: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).

% powr_less_mono
thf(fact_4839_powr__less__cancel,axiom,
    ! [X2: real,A2: real,B3: real] :
      ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ A2 @ B3 ) ) ) ).

% powr_less_cancel
thf(fact_4840_powr__mono,axiom,
    ! [A2: real,B3: real,X2: real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( ord_less_eq_real @ one_one_real @ X2 )
       => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).

% powr_mono
thf(fact_4841_sgn__not__eq__imp,axiom,
    ! [B3: int,A2: int] :
      ( ( ( sgn_sgn_int @ B3 )
       != ( sgn_sgn_int @ A2 ) )
     => ( ( ( sgn_sgn_int @ A2 )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B3 )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A2 )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B3 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_4842_sgn__not__eq__imp,axiom,
    ! [B3: real,A2: real] :
      ( ( ( sgn_sgn_real @ B3 )
       != ( sgn_sgn_real @ A2 ) )
     => ( ( ( sgn_sgn_real @ A2 )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B3 )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A2 )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B3 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_4843_sgn__not__eq__imp,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ( sgn_sgn_rat @ B3 )
       != ( sgn_sgn_rat @ A2 ) )
     => ( ( ( sgn_sgn_rat @ A2 )
         != zero_zero_rat )
       => ( ( ( sgn_sgn_rat @ B3 )
           != zero_zero_rat )
         => ( ( sgn_sgn_rat @ A2 )
            = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B3 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_4844_sgn__not__eq__imp,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B3 )
       != ( sgn_sgn_Code_integer @ A2 ) )
     => ( ( ( sgn_sgn_Code_integer @ A2 )
         != zero_z3403309356797280102nteger )
       => ( ( ( sgn_sgn_Code_integer @ B3 )
           != zero_z3403309356797280102nteger )
         => ( ( sgn_sgn_Code_integer @ A2 )
            = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B3 ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_4845_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_4846_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_4847_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_4848_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_4849_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_4850_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_4851_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_4852_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_4853_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_4854_powr__mono2_H,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ Y3 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).

% powr_mono2'
thf(fact_4855_powr__less__mono2,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y3 )
         => ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).

% powr_less_mono2
thf(fact_4856_gr__one__powr,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y3 ) ) ) ) ).

% gr_one_powr
thf(fact_4857_powr__inj,axiom,
    ! [A2: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ( powr_real @ A2 @ X2 )
            = ( powr_real @ A2 @ Y3 ) )
          = ( X2 = Y3 ) ) ) ) ).

% powr_inj
thf(fact_4858_ge__one__powr__ge__zero,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A2 ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_4859_powr__mono__both,axiom,
    ! [A2: real,B3: real,X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ A2 @ B3 )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ Y3 )
           => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ B3 ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_4860_powr__le1,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_4861_powr__divide,axiom,
    ! [X2: real,Y3: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( powr_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
          = ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).

% powr_divide
thf(fact_4862_powr__mult,axiom,
    ! [X2: real,Y3: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( powr_real @ ( times_times_real @ X2 @ Y3 ) @ A2 )
          = ( times_times_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).

% powr_mult
thf(fact_4863_sgn__1__pos,axiom,
    ! [A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A2 )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% sgn_1_pos
thf(fact_4864_sgn__1__pos,axiom,
    ! [A2: real] :
      ( ( ( sgn_sgn_real @ A2 )
        = one_one_real )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% sgn_1_pos
thf(fact_4865_sgn__1__pos,axiom,
    ! [A2: rat] :
      ( ( ( sgn_sgn_rat @ A2 )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% sgn_1_pos
thf(fact_4866_sgn__1__pos,axiom,
    ! [A2: int] :
      ( ( ( sgn_sgn_int @ A2 )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% sgn_1_pos
thf(fact_4867_sgn__root,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X2 ) )
        = ( sgn_sgn_real @ X2 ) ) ) ).

% sgn_root
thf(fact_4868_abs__sgn__eq,axiom,
    ! [A2: code_integer] :
      ( ( ( A2 = zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( A2 != zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
          = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
thf(fact_4869_abs__sgn__eq,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
          = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
          = one_one_real ) ) ) ).

% abs_sgn_eq
thf(fact_4870_abs__sgn__eq,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
          = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
          = one_one_rat ) ) ) ).

% abs_sgn_eq
thf(fact_4871_abs__sgn__eq,axiom,
    ! [A2: int] :
      ( ( ( A2 = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
          = zero_zero_int ) )
      & ( ( A2 != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_4872_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_4873_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_4874_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_4875_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_4876_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_4877_sgn__1__neg,axiom,
    ! [A2: int] :
      ( ( ( sgn_sgn_int @ A2 )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_4878_sgn__1__neg,axiom,
    ! [A2: real] :
      ( ( ( sgn_sgn_real @ A2 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% sgn_1_neg
thf(fact_4879_sgn__1__neg,axiom,
    ! [A2: rat] :
      ( ( ( sgn_sgn_rat @ A2 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_4880_sgn__1__neg,axiom,
    ! [A2: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_4881_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_4882_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_4883_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_4884_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_4885_powr__realpow,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X2 @ N ) ) ) ).

% powr_realpow
thf(fact_4886_powr__less__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( powr_real @ B3 @ Y3 ) @ X2 )
          = ( ord_less_real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).

% powr_less_iff
thf(fact_4887_less__powr__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y3 ) )
          = ( ord_less_real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).

% less_powr_iff
thf(fact_4888_log__less__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ B3 @ X2 ) @ Y3 )
          = ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ).

% log_less_iff
thf(fact_4889_less__log__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ Y3 @ ( log @ B3 @ X2 ) )
          = ( ord_less_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ).

% less_log_iff
thf(fact_4890_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_4891_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_4892_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_4893_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_4894_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_4895_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_4896_powr__neg__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).

% powr_neg_one
thf(fact_4897_powr__mult__base,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y3 ) )
        = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).

% powr_mult_base
thf(fact_4898_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X2: real,B3: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A2 ) @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
        = X2 )
     => ( ( X2
          = ( times_times_real @ ( sgn_sgn_real @ B3 ) @ ( power_power_real @ ( abs_abs_real @ B3 ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A2 = B3 ) ) ) ) ).

% sgn_power_injE
thf(fact_4899_le__log__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ Y3 @ ( log @ B3 @ X2 ) )
          = ( ord_less_eq_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ).

% le_log_iff
thf(fact_4900_log__le__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y3 )
          = ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ).

% log_le_iff
thf(fact_4901_le__powr__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y3 ) )
          = ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).

% le_powr_iff
thf(fact_4902_powr__le__iff,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( powr_real @ B3 @ Y3 ) @ X2 )
          = ( ord_less_eq_real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).

% powr_le_iff
thf(fact_4903_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q6: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q6 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q6 ) ) )
            & ( P @ Q6 ) ) ) ) ).

% split_div'
thf(fact_4904_ln__powr__bound,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ A2 ) ) ) ) ).

% ln_powr_bound
thf(fact_4905_ln__powr__bound2,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A2 ) @ ( times_times_real @ ( powr_real @ A2 @ A2 ) @ X2 ) ) ) ) ).

% ln_powr_bound2
thf(fact_4906_add__log__eq__powr,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ Y3 @ ( log @ B3 @ X2 ) )
            = ( log @ B3 @ ( times_times_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_4907_log__add__eq__powr,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ ( log @ B3 @ X2 ) @ Y3 )
            = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_4908_sgn__power__root,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X2 ) ) @ N ) )
        = X2 ) ) ).

% sgn_power_root
thf(fact_4909_root__sgn__power,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
        = Y3 ) ) ).

% root_sgn_power
thf(fact_4910_minus__log__eq__powr,axiom,
    ! [B3: real,X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ Y3 @ ( log @ B3 @ X2 ) )
            = ( log @ B3 @ ( divide_divide_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_4911_powr__def,axiom,
    ( powr_real
    = ( ^ [X: real,A4: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X ) ) ) ) ) ) ).

% powr_def
thf(fact_4912_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_4913_zero__le__sgn__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_sgn_iff
thf(fact_4914_sgn__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_4915_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_4916_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_4917_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_4918_sgn__zero,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_zero
thf(fact_4919_sgn__zero,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_zero
thf(fact_4920_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4921_mul__shift,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ( times_times_nat @ X2 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_4922_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_4923_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_4924_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_4925_sgn__zero__iff,axiom,
    ! [X2: complex] :
      ( ( ( sgn_sgn_complex @ X2 )
        = zero_zero_complex )
      = ( X2 = zero_zero_complex ) ) ).

% sgn_zero_iff
thf(fact_4926_sgn__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( sgn_sgn_real @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% sgn_zero_iff
thf(fact_4927_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q6: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q6 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q6: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q6 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_4928_eucl__rel__int_Ocases,axiom,
    ! [A13: int,A24: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A13 @ A24 @ A33 )
     => ( ( ( A24 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A13 ) ) )
       => ( ! [Q4: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
             => ( ( A24 != zero_zero_int )
               => ( A13
                 != ( times_times_int @ Q4 @ A24 ) ) ) )
         => ~ ! [R3: int,Q4: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q4 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A24 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A24 ) )
                   => ( A13
                     != ( plus_plus_int @ ( times_times_int @ Q4 @ A24 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_4929_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_4930_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_4931_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_4932_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_4933_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4934_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_4935_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4936_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_4937_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4938_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_4939_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_4940_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B3: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B3 @ X2 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
            & ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_4941_bezw__0,axiom,
    ! [X2: nat] :
      ( ( bezw @ X2 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_4942_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N: nat] :
      ( ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ ( set_list_VEBT_VEBT2 @ Xs ) )
         => ( ( size_s6755466524823107622T_VEBT @ X5 )
            = N ) )
     => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
        = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4943_length__mul__elem,axiom,
    ! [Xs: list_list_o,N: nat] :
      ( ! [X5: list_o] :
          ( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
         => ( ( size_size_list_o @ X5 )
            = N ) )
     => ( ( size_size_list_o @ ( concat_o @ Xs ) )
        = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4944_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
         => ( ( size_size_list_nat @ X5 )
            = N ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).

% length_mul_elem
thf(fact_4945_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A2 @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_4946_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A2 @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_4947_gbinomial__absorption_H,axiom,
    ! [K: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A2 @ K )
        = ( times_times_real @ ( divide_divide_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_4948_nth__Cons__pos,axiom,
    ! [N: nat,X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
        = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4949_nth__Cons__pos,axiom,
    ! [N: nat,X2: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4950_nth__Cons__pos,axiom,
    ! [N: nat,X2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_4951_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
     => ( ( rotate1_VEBT_VEBT @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4952_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
     => ( ( rotate1_o @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4953_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_4954_remove__def,axiom,
    ( remove6466555014256735590at_nat
    = ( ^ [X: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% remove_def
thf(fact_4955_remove__def,axiom,
    ( remove_real
    = ( ^ [X: real,A6: set_real] : ( minus_minus_set_real @ A6 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% remove_def
thf(fact_4956_remove__def,axiom,
    ( remove_o
    = ( ^ [X: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_4957_remove__def,axiom,
    ( remove_int
    = ( ^ [X: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% remove_def
thf(fact_4958_remove__def,axiom,
    ( remove_nat
    = ( ^ [X: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_4959_member__remove,axiom,
    ! [X2: real,Y3: real,A3: set_real] :
      ( ( member_real @ X2 @ ( remove_real @ Y3 @ A3 ) )
      = ( ( member_real @ X2 @ A3 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_4960_member__remove,axiom,
    ! [X2: $o,Y3: $o,A3: set_o] :
      ( ( member_o @ X2 @ ( remove_o @ Y3 @ A3 ) )
      = ( ( member_o @ X2 @ A3 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_4961_member__remove,axiom,
    ! [X2: set_nat,Y3: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X2 @ ( remove_set_nat @ Y3 @ A3 ) )
      = ( ( member_set_nat @ X2 @ A3 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_4962_member__remove,axiom,
    ! [X2: nat,Y3: nat,A3: set_nat] :
      ( ( member_nat @ X2 @ ( remove_nat @ Y3 @ A3 ) )
      = ( ( member_nat @ X2 @ A3 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_4963_member__remove,axiom,
    ! [X2: int,Y3: int,A3: set_int] :
      ( ( member_int @ X2 @ ( remove_int @ Y3 @ A3 ) )
      = ( ( member_int @ X2 @ A3 )
        & ( X2 != Y3 ) ) ) ).

% member_remove
thf(fact_4964_nth__Cons__0,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_4965_nth__Cons__0,axiom,
    ! [X2: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_4966_nth__Cons__0,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_4967_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_4968_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_4969_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_4970_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_4971_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4972_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4973_gbinomial__0_I1_J,axiom,
    ! [A2: complex] :
      ( ( gbinomial_complex @ A2 @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_4974_gbinomial__0_I1_J,axiom,
    ! [A2: real] :
      ( ( gbinomial_real @ A2 @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_4975_gbinomial__0_I1_J,axiom,
    ! [A2: rat] :
      ( ( gbinomial_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_4976_gbinomial__0_I1_J,axiom,
    ! [A2: nat] :
      ( ( gbinomial_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_4977_gbinomial__0_I1_J,axiom,
    ! [A2: int] :
      ( ( gbinomial_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_4978_ceiling__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_4979_ceiling__le__zero,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_4980_zero__less__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).

% zero_less_ceiling
thf(fact_4981_zero__less__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% zero_less_ceiling
thf(fact_4982_ceiling__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_4983_ceiling__less__one,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
      = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_4984_one__le__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).

% one_le_ceiling
thf(fact_4985_one__le__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% one_le_ceiling
thf(fact_4986_ceiling__le__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
      = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_4987_ceiling__le__one,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
      = ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_4988_one__less__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ one_one_rat @ X2 ) ) ).

% one_less_ceiling
thf(fact_4989_one__less__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ one_one_real @ X2 ) ) ).

% one_less_ceiling
thf(fact_4990_ceiling__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_4991_ceiling__less__zero,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_4992_zero__le__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 ) ) ).

% zero_le_ceiling
thf(fact_4993_zero__le__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 ) ) ).

% zero_le_ceiling
thf(fact_4994_ceiling__mono,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ Y3 @ X2 )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X2 ) ) ) ).

% ceiling_mono
thf(fact_4995_ceiling__mono,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).

% ceiling_mono
thf(fact_4996_ceiling__less__cancel,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
     => ( ord_less_rat @ X2 @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_4997_ceiling__less__cancel,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) )
     => ( ord_less_real @ X2 @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_4998_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_4999_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5000_set__subset__Cons,axiom,
    ! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5001_set__subset__Cons,axiom,
    ! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_5002_impossible__Cons,axiom,
    ! [Xs: list_int,Ys2: list_int,X2: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) )
     => ( Xs
       != ( cons_int @ X2 @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_5003_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( Xs
       != ( cons_VEBT_VEBT @ X2 @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_5004_impossible__Cons,axiom,
    ! [Xs: list_o,Ys2: list_o,X2: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) )
     => ( Xs
       != ( cons_o @ X2 @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_5005_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys2: list_nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
     => ( Xs
       != ( cons_nat @ X2 @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_5006_find_Osimps_I2_J,axiom,
    ! [P: int > $o,X2: int,Xs: list_int] :
      ( ( ( P @ X2 )
       => ( ( find_int @ P @ ( cons_int @ X2 @ Xs ) )
          = ( some_int @ X2 ) ) )
      & ( ~ ( P @ X2 )
       => ( ( find_int @ P @ ( cons_int @ X2 @ Xs ) )
          = ( find_int @ P @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5007_find_Osimps_I2_J,axiom,
    ! [P: nat > $o,X2: nat,Xs: list_nat] :
      ( ( ( P @ X2 )
       => ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
          = ( some_nat @ X2 ) ) )
      & ( ~ ( P @ X2 )
       => ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
          = ( find_nat @ P @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5008_find_Osimps_I2_J,axiom,
    ! [P: product_prod_nat_nat > $o,X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( P @ X2 )
       => ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
          = ( some_P7363390416028606310at_nat @ X2 ) ) )
      & ( ~ ( P @ X2 )
       => ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
          = ( find_P8199882355184865565at_nat @ P @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5009_find_Osimps_I2_J,axiom,
    ! [P: num > $o,X2: num,Xs: list_num] :
      ( ( ( P @ X2 )
       => ( ( find_num @ P @ ( cons_num @ X2 @ Xs ) )
          = ( some_num @ X2 ) ) )
      & ( ~ ( P @ X2 )
       => ( ( find_num @ P @ ( cons_num @ X2 @ Xs ) )
          = ( find_num @ P @ Xs ) ) ) ) ).

% find.simps(2)
thf(fact_5010_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A2: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A2 )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A2 @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_5011_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A2: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A2 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A2 @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_5012_ceiling__add__le,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_5013_ceiling__add__le,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_5014_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5015_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
      = ( ? [X: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs
              = ( cons_VEBT_VEBT @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5016_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) )
      = ( ? [X: $o,Ys3: list_o] :
            ( ( Xs
              = ( cons_o @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5017_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_5018_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A2 @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A2 @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_5019_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_5020_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A2: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A2 @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A2 @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_5021_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5022_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5023_list_Osize_I4_J,axiom,
    ! [X21: $o,X222: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_o @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5024_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_5025_nth__Cons_H,axiom,
    ! [N: nat,X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
          = X2 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
          = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5026_nth__Cons_H,axiom,
    ! [N: nat,X2: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
          = X2 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
          = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5027_nth__Cons_H,axiom,
    ! [N: nat,X2: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
          = X2 ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_5028_mult__ceiling__le,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5029_mult__ceiling__le,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5030_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A2 @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_5031_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A2 @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_5032_gbinomial__reduce__nat,axiom,
    ! [K: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A2 @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_5033_nth__equal__first__eq,axiom,
    ! [X2: real,Xs: list_real,N: nat] :
      ( ~ ( member_real @ X2 @ ( set_real2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( ( ( nth_real @ ( cons_real @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5034_nth__equal__first__eq,axiom,
    ! [X2: set_nat,Xs: list_set_nat,N: nat] :
      ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( ( ( nth_set_nat @ ( cons_set_nat @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5035_nth__equal__first__eq,axiom,
    ! [X2: int,Xs: list_int,N: nat] :
      ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5036_nth__equal__first__eq,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ~ ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5037_nth__equal__first__eq,axiom,
    ! [X2: $o,Xs: list_o,N: nat] :
      ( ~ ( member_o @ X2 @ ( set_o2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( ( ( nth_o @ ( cons_o @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5038_nth__equal__first__eq,axiom,
    ! [X2: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
            = X2 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_5039_nth__non__equal__first__eq,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
      ( ( X2 != Y3 )
     => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
          = Y3 )
        = ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5040_nth__non__equal__first__eq,axiom,
    ! [X2: int,Y3: int,Xs: list_int,N: nat] :
      ( ( X2 != Y3 )
     => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
          = Y3 )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5041_nth__non__equal__first__eq,axiom,
    ! [X2: nat,Y3: nat,Xs: list_nat,N: nat] :
      ( ( X2 != Y3 )
     => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
          = Y3 )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_5042_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_5043_ceiling__eq,axiom,
    ! [N: int,X2: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X2 )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_5044_ceiling__eq,axiom,
    ! [N: int,X2: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X2 )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_5045_ceiling__divide__lower,axiom,
    ! [Q3: rat,P6: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P6 ) ) ).

% ceiling_divide_lower
thf(fact_5046_ceiling__divide__lower,axiom,
    ! [Q3: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P6 ) ) ).

% ceiling_divide_lower
thf(fact_5047__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 )
          & ! [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
             => ( vEBT_invar_vebt @ X4 @ 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_5048_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_5049_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_5050_verit__le__mono__div__int,axiom,
    ! [A3: int,B2: int,N: int] :
      ( ( ord_less_int @ A3 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B2 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B2 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_5051_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_5052_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_5053_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_5054_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_5055_case4_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% case4(10)
thf(fact_5056_case4_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList2 )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% case4(4)
thf(fact_5057_pow__sum,axiom,
    ! [A2: nat,B3: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ).

% pow_sum
thf(fact_5058_a0,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% a0
thf(fact_5059_case4_I7_J,axiom,
    ! [I3: nat] :
      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ X8 ) )
        = ( vEBT_V8194947554948674370ptions @ summary2 @ I3 ) ) ) ).

% case4(7)
thf(fact_5060_power__minus__is__div,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A2 @ B3 ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% power_minus_is_div
thf(fact_5061_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X2 )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_5062_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_5063_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_5064_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_5065_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_5066_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_5067_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_5068_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5069_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5070_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5071_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5072_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5073_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5074_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_5075_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_5076_bits__mod__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_5077_bits__mod__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_5078_mod__self,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ A2 )
      = zero_zero_int ) ).

% mod_self
thf(fact_5079_mod__self,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% mod_self
thf(fact_5080_mod__by__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ zero_zero_int )
      = A2 ) ).

% mod_by_0
thf(fact_5081_mod__by__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% mod_by_0
thf(fact_5082_mod__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mod_0
thf(fact_5083_mod__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mod_0
thf(fact_5084_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_5085_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = ( some_nat @ Y3 ) )
       => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_5086_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = ( some_nat @ Y3 ) )
       => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_5087_delt__out__of__range,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
        | ( ord_less_nat @ Ma @ X2 ) )
     => ( ( 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 ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_5088_del__single__cont,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( X2 = 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 ) @ X2 )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_5089_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_5090_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_5091_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_5092_succ__min,axiom,
    ! [Deg: nat,X2: 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 @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_5093_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_5094_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5095_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5096_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5097_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5098_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5099_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5100_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5101_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5102_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_5103_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_5104_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_5105_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_5106_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_5107_mod__mult__self1__is__0,axiom,
    ! [B3: int,A2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B3 @ A2 ) @ B3 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_5108_mod__mult__self1__is__0,axiom,
    ! [B3: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B3 @ A2 ) @ B3 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_5109_mod__mult__self2__is__0,axiom,
    ! [A2: int,B3: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B3 ) @ B3 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_5110_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B3 ) @ B3 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_5111_bits__mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_5112_bits__mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_5113_mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_5114_mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_5115_bits__mod__div__trivial,axiom,
    ! [A2: int,B3: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_5116_bits__mod__div__trivial,axiom,
    ! [A2: nat,B3: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_5117_mod__div__trivial,axiom,
    ! [A2: int,B3: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_5118_mod__div__trivial,axiom,
    ! [A2: nat,B3: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_5119_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = zero_zero_int )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5120_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = zero_zero_real )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5121_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = zero_zero_rat )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5122_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5123_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5124_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5125_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_5126_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_5127_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_5128_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_5129_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_5130_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W2 @ Z ) ) ).

% of_int_le_iff
thf(fact_5131_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_5132_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_5133_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W2 @ Z ) ) ).

% of_int_less_iff
thf(fact_5134_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_5135_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
      = zero_zero_rat ) ).

% frac_of_int
thf(fact_5136_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_5137_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_5138_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_5139_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_5140_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_5141_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_5142_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_5143_divide__le__eq__numeral1_I1_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A2 )
      = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_5144_divide__le__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A2 )
      = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_5145_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_5146_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_5147_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( A2
        = ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) )
            = B3 ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_5148_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( A2
        = ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) )
            = B3 ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_5149_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: complex,B3: complex,W2: num] :
      ( ( A2
        = ( divide1717551699836669952omplex @ B3 @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W2 ) )
            = B3 ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_5150_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) )
        = A2 )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_5151_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) )
        = A2 )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( B3
            = ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_5152_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: complex,W2: num,A2: complex] :
      ( ( ( divide1717551699836669952omplex @ B3 @ ( numera6690914467698888265omplex @ W2 ) )
        = A2 )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W2 ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_5153_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_5154_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_5155_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_5156_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_5157_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_5158_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_5159_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_5160_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_5161_divide__less__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A2 )
      = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_5162_divide__less__eq__numeral1_I1_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A2 )
      = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_5163_mod__minus1__right,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5164_mod__minus1__right,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_5165_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5166_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5167_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5168_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5169_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5170_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5171_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5172_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5173_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5174_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5175_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5176_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5177_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5178_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5179_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5180_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5181_ceiling__le__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5182_ceiling__le__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5183_numeral__less__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).

% numeral_less_ceiling
thf(fact_5184_numeral__less__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).

% numeral_less_ceiling
thf(fact_5185_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_5186_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_5187_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5188_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5189_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5190_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5191_divide__le__eq__numeral1_I2_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A2 )
      = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5192_divide__le__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A2 )
      = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5193_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5194_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5195_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( A2
        = ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
            = B3 ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5196_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( A2
        = ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
            = B3 ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5197_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: complex,B3: complex,W2: num] :
      ( ( A2
        = ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
            = B3 ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5198_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        = A2 )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( B3
            = ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5199_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
        = A2 )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5200_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: complex,W2: num,A2: complex] :
      ( ( ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
        = A2 )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5201_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5202_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5203_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5204_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5205_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B3: real,W2: num] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5206_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B3: rat,W2: num] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5207_divide__less__eq__numeral1_I2_J,axiom,
    ! [B3: real,W2: num,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A2 )
      = ( ord_less_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5208_divide__less__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W2: num,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A2 )
      = ( ord_less_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5209_zero__eq__power2,axiom,
    ! [A2: rat] :
      ( ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_5210_zero__eq__power2,axiom,
    ! [A2: int] :
      ( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_5211_zero__eq__power2,axiom,
    ! [A2: nat] :
      ( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_5212_zero__eq__power2,axiom,
    ! [A2: real] :
      ( ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_5213_zero__eq__power2,axiom,
    ! [A2: complex] :
      ( ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_5214_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5215_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5216_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5217_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5218_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5219_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5220_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5221_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5222_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5223_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5224_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5225_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5226_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5227_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5228_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5229_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5230_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5231_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5232_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5233_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5234_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5235_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5236_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5237_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5238_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5239_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5240_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5241_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5242_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5243_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5244_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5245_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5246_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W2: nat,X2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5247_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5248_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5249_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5250_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_5251_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_5252_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_5253_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_5254_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_5255_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_5256_power2__eq__iff__nonneg,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5257_power2__eq__iff__nonneg,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5258_power2__eq__iff__nonneg,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5259_power2__eq__iff__nonneg,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_5260_power2__less__eq__zero__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_5261_power2__less__eq__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_5262_power2__less__eq__zero__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_5263_zero__less__power2,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_5264_zero__less__power2,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_5265_zero__less__power2,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_5266_sum__power2__eq__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_5267_sum__power2__eq__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_5268_sum__power2__eq__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_5269_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_5270_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_5271_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_5272_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_5273_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_5274_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_5275_ceiling__less__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_5276_ceiling__less__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_5277_numeral__le__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).

% numeral_le_ceiling
thf(fact_5278_numeral__le__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).

% numeral_le_ceiling
thf(fact_5279_ceiling__le__neg__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5280_ceiling__le__neg__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5281_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5282_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5283_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5284_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5285_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5286_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5287_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5288_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5289_neg__numeral__less__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).

% neg_numeral_less_ceiling
thf(fact_5290_neg__numeral__less__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).

% neg_numeral_less_ceiling
thf(fact_5291_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5292_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5293_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5294_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5295_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5296_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5297_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5298_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5299_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5300_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5301_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5302_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5303_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5304_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5305_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5306_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5307_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5308_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5309_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5310_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5311_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5312_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5313_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5314_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5315_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5316_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5317_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5318_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5319_ceiling__less__neg__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_5320_ceiling__less__neg__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_5321_neg__numeral__le__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).

% neg_numeral_le_ceiling
thf(fact_5322_neg__numeral__le__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).

% neg_numeral_le_ceiling
thf(fact_5323_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5324_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5325_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5326_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5327_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5328_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5329_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5330_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5331_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5332_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5333_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5334_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5335_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5336_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5337_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5338_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5339_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 )
    & ! [X4: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ( vEBT_invar_vebt @ X4 @ na ) ) ) ).

% sprop1
thf(fact_5340_divmod__digit__0_I2_J,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
          = ( modulo364778990260209775nteger @ A2 @ B3 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5341_divmod__digit__0_I2_J,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
          = ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5342_divmod__digit__0_I2_J,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
          = ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5343_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_5344_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_5345_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_5346_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_5347_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_5348_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_5349_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_5350_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_5351_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_5352_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_5353_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_5354_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5355_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5356_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5357_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_5358_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_5359_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_5360_card__2__iff_H,axiom,
    ! [S: set_complex] :
      ( ( ( finite_card_complex @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: complex] :
            ( ( member_complex @ X @ S )
            & ? [Y: complex] :
                ( ( member_complex @ Y @ S )
                & ( X != Y )
                & ! [Z2: complex] :
                    ( ( member_complex @ Z2 @ S )
                   => ( ( Z2 = X )
                      | ( Z2 = Y ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5361_card__2__iff_H,axiom,
    ! [S: set_list_nat] :
      ( ( ( finite_card_list_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: list_nat] :
            ( ( member_list_nat @ X @ S )
            & ? [Y: list_nat] :
                ( ( member_list_nat @ Y @ S )
                & ( X != Y )
                & ! [Z2: list_nat] :
                    ( ( member_list_nat @ Z2 @ S )
                   => ( ( Z2 = X )
                      | ( Z2 = Y ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5362_card__2__iff_H,axiom,
    ! [S: set_set_nat] :
      ( ( ( finite_card_set_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: set_nat] :
            ( ( member_set_nat @ X @ S )
            & ? [Y: set_nat] :
                ( ( member_set_nat @ Y @ S )
                & ( X != Y )
                & ! [Z2: set_nat] :
                    ( ( member_set_nat @ Z2 @ S )
                   => ( ( Z2 = X )
                      | ( Z2 = Y ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5363_card__2__iff_H,axiom,
    ! [S: set_nat] :
      ( ( ( finite_card_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ S )
            & ? [Y: nat] :
                ( ( member_nat @ Y @ S )
                & ( X != Y )
                & ! [Z2: nat] :
                    ( ( member_nat @ Z2 @ S )
                   => ( ( Z2 = X )
                      | ( Z2 = Y ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5364_card__2__iff_H,axiom,
    ! [S: set_int] :
      ( ( ( finite_card_int @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: int] :
            ( ( member_int @ X @ S )
            & ? [Y: int] :
                ( ( member_int @ Y @ S )
                & ( X != Y )
                & ! [Z2: int] :
                    ( ( member_int @ Z2 @ S )
                   => ( ( Z2 = X )
                      | ( Z2 = Y ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5365_le__num__One__iff,axiom,
    ! [X2: num] :
      ( ( ord_less_eq_num @ X2 @ one )
      = ( X2 = one ) ) ).

% le_num_One_iff
thf(fact_5366_bits__stable__imp__add__self,axiom,
    ! [A2: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_p5714425477246183910nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_5367_bits__stable__imp__add__self,axiom,
    ! [A2: int] :
      ( ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_plus_int @ A2 @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_5368_bits__stable__imp__add__self,axiom,
    ! [A2: nat] :
      ( ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_plus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_5369_divmod__digit__0_I1_J,axiom,
    ! [B3: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
          = ( divide6298287555418463151nteger @ A2 @ B3 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5370_divmod__digit__0_I1_J,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
          = ( divide_divide_int @ A2 @ B3 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5371_divmod__digit__0_I1_J,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
          = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5372_sum__squares__bound,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_5373_sum__squares__bound,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_5374_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5375_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5376_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5377_half__gt__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_5378_half__gt__zero,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_5379_half__gt__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% half_gt_zero_iff
thf(fact_5380_half__gt__zero__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% half_gt_zero_iff
thf(fact_5381_field__less__half__sum,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ X2 @ Y3 )
     => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_5382_field__less__half__sum,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_5383_power2__le__imp__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5384_power2__le__imp__le,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5385_power2__le__imp__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5386_power2__le__imp__le,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_5387_power2__eq__imp__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5388_power2__eq__imp__eq,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5389_power2__eq__imp__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5390_power2__eq__imp__eq,axiom,
    ! [X2: int,Y3: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( X2 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_5391_zero__le__power2,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_5392_zero__le__power2,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_5393_zero__le__power2,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_5394_power2__less__0,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_5395_power2__less__0,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_5396_power2__less__0,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_5397_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5398_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5399_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5400_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_5401_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_5402_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_5403_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_5404_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_5405_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_5406_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_5407_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_5408_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_5409_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_5410_abs__le__square__iff,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y3 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5411_abs__le__square__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) )
      = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5412_abs__le__square__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y3 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5413_abs__le__square__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5414_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_5415_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_5416_card__2__iff,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
            ( ( S
              = ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5417_card__2__iff,axiom,
    ! [S: set_complex] :
      ( ( ( finite_card_complex @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: complex,Y: complex] :
            ( ( S
              = ( insert_complex @ X @ ( insert_complex @ Y @ bot_bot_set_complex ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5418_card__2__iff,axiom,
    ! [S: set_list_nat] :
      ( ( ( finite_card_list_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: list_nat,Y: list_nat] :
            ( ( S
              = ( insert_list_nat @ X @ ( insert_list_nat @ Y @ bot_bot_set_list_nat ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5419_card__2__iff,axiom,
    ! [S: set_set_nat] :
      ( ( ( finite_card_set_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: set_nat,Y: set_nat] :
            ( ( S
              = ( insert_set_nat @ X @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5420_card__2__iff,axiom,
    ! [S: set_real] :
      ( ( ( finite_card_real @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: real,Y: real] :
            ( ( S
              = ( insert_real @ X @ ( insert_real @ Y @ bot_bot_set_real ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5421_card__2__iff,axiom,
    ! [S: set_o] :
      ( ( ( finite_card_o @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: $o,Y: $o] :
            ( ( S
              = ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5422_card__2__iff,axiom,
    ! [S: set_nat] :
      ( ( ( finite_card_nat @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: nat,Y: nat] :
            ( ( S
              = ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5423_card__2__iff,axiom,
    ! [S: set_int] :
      ( ( ( finite_card_int @ S )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X: int,Y: int] :
            ( ( S
              = ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) )
            & ( X != Y ) ) ) ) ).

% card_2_iff
thf(fact_5424_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_5425_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_5426_realpow__square__minus__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_5427_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_5428_ex__le__of__int,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_le_of_int
thf(fact_5429_ex__le__of__int,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_le_of_int
thf(fact_5430_ex__less__of__int,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_less_of_int
thf(fact_5431_ex__less__of__int,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_less_of_int
thf(fact_5432_ex__of__int__less,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X2 ) ).

% ex_of_int_less
thf(fact_5433_ex__of__int__less,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 ) ).

% ex_of_int_less
thf(fact_5434_mod__double__modulus,axiom,
    ! [M: code_integer,X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
       => ( ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X2 @ M ) )
          | ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5435_mod__double__modulus,axiom,
    ! [M: nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
       => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X2 @ M ) )
          | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5436_mod__double__modulus,axiom,
    ! [M: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X2 @ M ) )
          | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5437_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
            = ( modulo364778990260209775nteger @ A2 @ B3 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5438_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
            = ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5439_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
            = ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5440_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_5441_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_5442_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_5443_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_5444_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N ) ) ).

% zero_neq_numeral
thf(fact_5445_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( numera6620942414471956472nteger @ N ) ) ).

% zero_neq_numeral
thf(fact_5446_power2__less__imp__less,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ X2 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5447_power2__less__imp__less,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ X2 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5448_power2__less__imp__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_nat @ X2 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5449_power2__less__imp__less,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_int @ X2 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_5450_sum__power2__ge__zero,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_5451_sum__power2__ge__zero,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_5452_sum__power2__ge__zero,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_5453_sum__power2__le__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_5454_sum__power2__le__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_5455_sum__power2__le__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_5456_sum__power2__gt__zero__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_5457_sum__power2__gt__zero__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_5458_sum__power2__gt__zero__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_5459_not__sum__power2__lt__zero,axiom,
    ! [X2: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_5460_not__sum__power2__lt__zero,axiom,
    ! [X2: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_5461_not__sum__power2__lt__zero,axiom,
    ! [X2: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_5462_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_5463_square__le__1,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_5464_square__le__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
     => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5465_square__le__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5466_square__le__1,axiom,
    ! [X2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
     => ( ( ord_less_eq_int @ X2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_5467_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5468_power2__le__iff__abs__le,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5469_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5470_power2__le__iff__abs__le,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5471_zero__le__even__power_H,axiom,
    ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_5472_zero__le__even__power_H,axiom,
    ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_5473_zero__le__even__power_H,axiom,
    ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_5474_abs__square__le__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_5475_abs__square__le__1,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_5476_abs__square__le__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_5477_abs__square__le__1,axiom,
    ! [X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_5478_abs__square__less__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_5479_abs__square__less__1,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_5480_abs__square__less__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_5481_abs__square__less__1,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_5482_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_5483_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_5484_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_5485_arith__geo__mean,axiom,
    ! [U: real,X2: real,Y3: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X2 @ Y3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_5486_arith__geo__mean,axiom,
    ! [U: rat,X2: rat,Y3: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X2 @ Y3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_5487_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A2 @ B3 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5488_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5489_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A2 @ B3 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5490_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_5491_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_5492_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_5493_odd__power__less__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_5494_odd__power__less__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_5495_odd__power__less__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_5496_ex__power__ivl2,axiom,
    ! [B3: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B3 @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_5497_ex__power__ivl1,axiom,
    ! [B3: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_5498_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ A2 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5499_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B3 ) @ A2 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_5500_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ord_less_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5501_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_5502_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: int,B3: int] :
      ( ( ( modulo_modulo_int @ A2 @ B3 )
        = A2 )
      = ( ( divide_divide_int @ A2 @ B3 )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_5503_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ B3 )
        = A2 )
      = ( ( divide_divide_nat @ A2 @ B3 )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_5504_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_5505_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_le_numeral
thf(fact_5506_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_le_numeral
thf(fact_5507_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_5508_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_5509_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_5510_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_5511_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_5512_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_le_zero
thf(fact_5513_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_5514_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_5515_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_5516_exp__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_5517_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_5518_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_5519_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_5520_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_5521_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_5522_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_less_zero
thf(fact_5523_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_5524_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_5525_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_5526_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_5527_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_less_numeral
thf(fact_5528_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_less_numeral
thf(fact_5529_le__of__int__ceiling,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ).

% le_of_int_ceiling
thf(fact_5530_le__of__int__ceiling,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).

% le_of_int_ceiling
thf(fact_5531_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_5532_one__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% one_le_numeral
thf(fact_5533_one__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).

% one_le_numeral
thf(fact_5534_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_5535_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_5536_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_5537_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_5538_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_5539_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_5540_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_5541_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_5542_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_5543_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5544_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5545_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5546_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5547_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5548_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5549_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5550_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5551_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5552_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5553_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5554_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5555_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5556_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_5557_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5558_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5559_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5560_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5561_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5562_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5563_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5564_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5565_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_5566_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_5567_ln__one__plus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_5568_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_5569_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_5570_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A2 @ B3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5571_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5572_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ B3 )
       => ( ( modulo_modulo_nat @ A2 @ B3 )
          = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5573_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ B3 )
       => ( ( modulo_modulo_int @ A2 @ B3 )
          = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5574_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_5575_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_5576_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5577_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_5578_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_5579_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_5580_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5581_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_5582_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_5583_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_5584_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_5585_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_5586_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5587_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_5588_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_5589_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_5590_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5591_ceiling__le,axiom,
    ! [X2: real,A2: int] :
      ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A2 ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A2 ) ) ).

% ceiling_le
thf(fact_5592_ceiling__le,axiom,
    ! [X2: rat,A2: int] :
      ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A2 ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A2 ) ) ).

% ceiling_le
thf(fact_5593_ceiling__le__iff,axiom,
    ! [X2: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
      = ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5594_ceiling__le__iff,axiom,
    ! [X2: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
      = ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5595_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W2 )
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_5596_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_5597_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W2 )
        = ( divide1717551699836669952omplex @ B3 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_5598_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B3 @ C )
        = ( numeral_numeral_rat @ W2 ) )
      = ( ( ( C != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_5599_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B3 @ C )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C != zero_zero_real )
         => ( B3
            = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_5600_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B3 @ C )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( ( C != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_5601_less__ceiling__iff,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).

% less_ceiling_iff
thf(fact_5602_less__ceiling__iff,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).

% less_ceiling_iff
thf(fact_5603_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5604_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5605_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5606_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5607_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_5608_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5609_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_5610_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_5611_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_5612_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_5613_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_5614_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_5615_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_5616_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5617_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_5618_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_5619_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_5620_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5621_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_5622_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_5623_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_5624_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_5625_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_5626_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5627_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_5628_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_5629_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_5630_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5631_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_5632_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_5633_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_5634_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5635_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5636_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5637_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5638_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5639_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5640_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5641_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5642_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5643_neg__mod__conj,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B3 ) @ zero_zero_int )
        & ( ord_less_int @ B3 @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).

% neg_mod_conj
thf(fact_5644_pos__mod__conj,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B3 ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 ) ) ) ).

% pos_mod_conj
thf(fact_5645_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_5646_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_5647_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_5648_real__of__int__div4,axiom,
    ! [N: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) ) ).

% real_of_int_div4
thf(fact_5649_zdiv__mono__strict,axiom,
    ! [A3: int,B2: int,N: int] :
      ( ( ord_less_int @ A3 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A3 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B2 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_5650_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_5651_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_5652_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5653_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5654_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5655_of__int__leD,axiom,
    ! [N: int,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).

% of_int_leD
thf(fact_5656_of__int__leD,axiom,
    ! [N: int,X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% of_int_leD
thf(fact_5657_of__int__leD,axiom,
    ! [N: int,X2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).

% of_int_leD
thf(fact_5658_of__int__leD,axiom,
    ! [N: int,X2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).

% of_int_leD
thf(fact_5659_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_5660_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_5661_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_5662_of__int__lessD,axiom,
    ! [N: int,X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).

% of_int_lessD
thf(fact_5663_of__int__lessD,axiom,
    ! [N: int,X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% of_int_lessD
thf(fact_5664_of__int__lessD,axiom,
    ! [N: int,X2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).

% of_int_lessD
thf(fact_5665_of__int__lessD,axiom,
    ! [N: int,X2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X2 ) ) ) ).

% of_int_lessD
thf(fact_5666_floor__exists1,axiom,
    ! [X2: real] :
    ? [X5: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X2 )
            & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
thf(fact_5667_floor__exists1,axiom,
    ! [X2: rat] :
    ? [X5: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X2 )
            & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
thf(fact_5668_floor__exists,axiom,
    ! [X2: real] :
    ? [Z4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5669_floor__exists,axiom,
    ! [X2: rat] :
    ? [Z4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5670_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_5671_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_5672_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_5673_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_5674_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_5675_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_5676_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X2: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_5677_ceiling__log__nat__eq__if,axiom,
    ! [B3: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_5678_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_5679_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_5680_divide__less__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_5681_divide__less__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_5682_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5683_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5684_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B3 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5685_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B3 @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B3
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5686_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B3 @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5687_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B3 @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5688_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_5689_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).

% int_less_real_le
thf(fact_5690_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_5691_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_5692_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_5693_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ B3 @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5694_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B3 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5695_ceiling__correct,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) @ one_one_real ) @ X2 )
      & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).

% ceiling_correct
thf(fact_5696_ceiling__correct,axiom,
    ! [X2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) @ one_one_rat ) @ X2 )
      & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ) ).

% ceiling_correct
thf(fact_5697_ceiling__unique,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X2 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5698_ceiling__unique,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X2 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5699_ceiling__eq__iff,axiom,
    ! [X2: real,A2: int] :
      ( ( ( archim7802044766580827645g_real @ X2 )
        = A2 )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A2 ) @ one_one_real ) @ X2 )
        & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A2 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5700_ceiling__eq__iff,axiom,
    ! [X2: rat,A2: int] :
      ( ( ( archim2889992004027027881ng_rat @ X2 )
        = A2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) @ X2 )
        & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A2 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5701_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_5702_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_5703_ceiling__less__iff,axiom,
    ! [X2: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
      = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_5704_ceiling__less__iff,axiom,
    ! [X2: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
      = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_5705_le__ceiling__iff,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).

% le_ceiling_iff
thf(fact_5706_le__ceiling__iff,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).

% le_ceiling_iff
thf(fact_5707_divide__le__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_5708_divide__le__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_5709_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_5710_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_5711_divide__less__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5712_divide__less__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5713_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5714_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5715_real__of__int__div2,axiom,
    ! [N: int,X2: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) ) ).

% real_of_int_div2
thf(fact_5716_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_5717_int__mod__neg__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B3 @ R2 )
         => ( ( modulo_modulo_int @ A2 @ B3 )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_5718_int__mod__pos__eq,axiom,
    ! [A2: int,B3: int,Q3: int,R2: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B3 )
         => ( ( modulo_modulo_int @ A2 @ B3 )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_5719_real__of__int__div3,axiom,
    ! [N: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_5720_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5721_zmod__minus1,axiom,
    ! [B3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
        = ( minus_minus_int @ B3 @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_5722_zmod__zmult2__eq,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B3 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_5723_ceiling__divide__upper,axiom,
    ! [Q3: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_5724_ceiling__divide__upper,axiom,
    ! [Q3: rat,P6: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_5725_mult__ceiling__le__Ints,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( member_real @ A2 @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5726_mult__ceiling__le__Ints,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( member_real @ A2 @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5727_mult__ceiling__le__Ints,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( member_real @ A2 @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5728_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5729_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5730_mult__ceiling__le__Ints,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( member_rat @ A2 @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_5731_divide__le__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5732_divide__le__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5733_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5734_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5735_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_5736_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_5737_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_5738_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y3 )
           => ( ( vEBT_vebt_member @ T @ Y3 )
              | ( X2 = Y3 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_5739_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X2 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_5740_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_5741_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = 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 ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_5742_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T @ X2 ) @ N ) ) ) ).

% valid_pres_insert
thf(fact_5743_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_5744_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_5745_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_5746_Icc__eq__Icc,axiom,
    ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H2 )
        = ( set_or370866239135849197et_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H2 )
          & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5747_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5748_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5749_Icc__eq__Icc,axiom,
    ! [L: nat,H2: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H2 )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5750_Icc__eq__Icc,axiom,
    ! [L: int,H2: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H2 )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H2 )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5751_Icc__eq__Icc,axiom,
    ! [L: real,H2: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H2 )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L @ H2 )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5752_atLeastAtMost__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_eq_o @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5753_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5754_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I )
        & ( ord_less_eq_set_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5755_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5756_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5757_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5758_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5759_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5760_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_5761_atLeastatMost__empty__iff2,axiom,
    ! [A2: $o,B3: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_o @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5762_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5763_atLeastatMost__empty__iff2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5764_atLeastatMost__empty__iff2,axiom,
    ! [A2: num,B3: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5765_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5766_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B3: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5767_atLeastatMost__empty__iff2,axiom,
    ! [A2: real,B3: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A2 @ B3 ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5768_atLeastatMost__empty__iff,axiom,
    ! [A2: $o,B3: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5769_atLeastatMost__empty__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ( set_or370866239135849197et_int @ A2 @ B3 )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5770_atLeastatMost__empty__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ( set_or633870826150836451st_rat @ A2 @ B3 )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5771_atLeastatMost__empty__iff,axiom,
    ! [A2: num,B3: num] :
      ( ( ( set_or7049704709247886629st_num @ A2 @ B3 )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5772_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5773_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B3: int] :
      ( ( ( set_or1266510415728281911st_int @ A2 @ B3 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5774_atLeastatMost__empty__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ( set_or1222579329274155063t_real @ A2 @ B3 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5775_atLeastatMost__subset__iff,axiom,
    ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A2 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
        | ( ( ord_less_eq_set_int @ C @ A2 )
          & ( ord_less_eq_set_int @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5776_atLeastatMost__subset__iff,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B3 )
        | ( ( ord_less_eq_rat @ C @ A2 )
          & ( ord_less_eq_rat @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5777_atLeastatMost__subset__iff,axiom,
    ! [A2: num,B3: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A2 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B3 )
        | ( ( ord_less_eq_num @ C @ A2 )
          & ( ord_less_eq_num @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5778_atLeastatMost__subset__iff,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
        | ( ( ord_less_eq_nat @ C @ A2 )
          & ( ord_less_eq_nat @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5779_atLeastatMost__subset__iff,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B3 )
        | ( ( ord_less_eq_int @ C @ A2 )
          & ( ord_less_eq_int @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5780_atLeastatMost__subset__iff,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B3 )
        | ( ( ord_less_eq_real @ C @ A2 )
          & ( ord_less_eq_real @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5781_atLeastatMost__empty,axiom,
    ! [B3: $o,A2: $o] :
      ( ( ord_less_o @ B3 @ A2 )
     => ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_5782_atLeastatMost__empty,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( set_or633870826150836451st_rat @ A2 @ B3 )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_5783_atLeastatMost__empty,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ( ( set_or7049704709247886629st_num @ A2 @ B3 )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_5784_atLeastatMost__empty,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_5785_atLeastatMost__empty,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( set_or1266510415728281911st_int @ A2 @ B3 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_5786_atLeastatMost__empty,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( set_or1222579329274155063t_real @ A2 @ B3 )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_5787_infinite__Icc__iff,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) ) )
      = ( ord_less_rat @ A2 @ B3 ) ) ).

% infinite_Icc_iff
thf(fact_5788_infinite__Icc__iff,axiom,
    ! [A2: real,B3: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) ) )
      = ( ord_less_real @ A2 @ B3 ) ) ).

% infinite_Icc_iff
thf(fact_5789_atLeastAtMost__singleton,axiom,
    ! [A2: $o] :
      ( ( set_or8904488021354931149Most_o @ A2 @ A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_5790_atLeastAtMost__singleton,axiom,
    ! [A2: nat] :
      ( ( set_or1269000886237332187st_nat @ A2 @ A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_5791_atLeastAtMost__singleton,axiom,
    ! [A2: int] :
      ( ( set_or1266510415728281911st_int @ A2 @ A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_5792_atLeastAtMost__singleton,axiom,
    ! [A2: real] :
      ( ( set_or1222579329274155063t_real @ A2 @ A2 )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_5793_atLeastAtMost__singleton__iff,axiom,
    ! [A2: $o,B3: $o,C: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
        = ( insert_o @ C @ bot_bot_set_o ) )
      = ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5794_atLeastAtMost__singleton__iff,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5795_atLeastAtMost__singleton__iff,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A2 @ B3 )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5796_atLeastAtMost__singleton__iff,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A2 @ B3 )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A2 = B3 )
        & ( B3 = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5797_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W2: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_5798_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_5799_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_5800_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_5801_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_5802_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_5803_powr__numeral,axiom,
    ! [X2: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_5804_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_5805_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_5806_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_5807_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_5808_infinite__Icc,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) ) ) ).

% infinite_Icc
thf(fact_5809_infinite__Icc,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) ) ) ).

% infinite_Icc
thf(fact_5810_atLeastAtMost__singleton_H,axiom,
    ! [A2: $o,B3: $o] :
      ( ( A2 = B3 )
     => ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
        = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5811_atLeastAtMost__singleton_H,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 = B3 )
     => ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
        = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5812_atLeastAtMost__singleton_H,axiom,
    ! [A2: int,B3: int] :
      ( ( A2 = B3 )
     => ( ( set_or1266510415728281911st_int @ A2 @ B3 )
        = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5813_atLeastAtMost__singleton_H,axiom,
    ! [A2: real,B3: real] :
      ( ( A2 = B3 )
     => ( ( set_or1222579329274155063t_real @ A2 @ B3 )
        = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5814_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_5815_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N )
           => ( P @ M2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_5816_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N )
            & ( P @ M2 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_5817_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P6: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P6 )
       => ( ( ord_less_nat @ M @ P6 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P6 )
               => ( ( P @ N3 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P6 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_5818_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_5819_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
             => ( P @ M4 @ N3 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_5820_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_5821_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q4: nat] :
          ( M
          = ( times_times_nat @ D @ Q4 ) ) ) ).

% mod_eq_0D
thf(fact_5822_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_5823_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_5824_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_5825_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X2 )
      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(2)
thf(fact_5826_atLeastatMost__psubset__iff,axiom,
    ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A2 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
          | ( ( ord_less_eq_set_int @ C @ A2 )
            & ( ord_less_eq_set_int @ B3 @ D )
            & ( ( ord_less_set_int @ C @ A2 )
              | ( ord_less_set_int @ B3 @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5827_atLeastatMost__psubset__iff,axiom,
    ! [A2: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A2 @ B3 )
          | ( ( ord_less_eq_rat @ C @ A2 )
            & ( ord_less_eq_rat @ B3 @ D )
            & ( ( ord_less_rat @ C @ A2 )
              | ( ord_less_rat @ B3 @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5828_atLeastatMost__psubset__iff,axiom,
    ! [A2: num,B3: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A2 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A2 @ B3 )
          | ( ( ord_less_eq_num @ C @ A2 )
            & ( ord_less_eq_num @ B3 @ D )
            & ( ( ord_less_num @ C @ A2 )
              | ( ord_less_num @ B3 @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5829_atLeastatMost__psubset__iff,axiom,
    ! [A2: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A2 @ B3 )
          | ( ( ord_less_eq_nat @ C @ A2 )
            & ( ord_less_eq_nat @ B3 @ D )
            & ( ( ord_less_nat @ C @ A2 )
              | ( ord_less_nat @ B3 @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5830_atLeastatMost__psubset__iff,axiom,
    ! [A2: int,B3: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A2 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A2 @ B3 )
          | ( ( ord_less_eq_int @ C @ A2 )
            & ( ord_less_eq_int @ B3 @ D )
            & ( ( ord_less_int @ C @ A2 )
              | ( ord_less_int @ B3 @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5831_atLeastatMost__psubset__iff,axiom,
    ! [A2: real,B3: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A2 @ B3 )
          | ( ( ord_less_eq_real @ C @ A2 )
            & ( ord_less_eq_real @ B3 @ D )
            & ( ( ord_less_real @ C @ A2 )
              | ( ord_less_real @ B3 @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5832_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_5833_div__less__mono,axiom,
    ! [A3: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A3 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B2 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_5834_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_5835_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S3: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_5836_nat__mod__eq__lemma,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X2 @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
     => ( ( ord_less_eq_nat @ Y3 @ X2 )
       => ? [Q4: nat] :
            ( X2
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q4 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_5837_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_5838_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_5839_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_5840_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_5841_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_5842_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X2 )
      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(3)
thf(fact_5843_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A2: $o,B3: $o] :
      ( ( ( X2 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
          = ( vEBT_Leaf @ $true @ B3 ) ) )
      & ( ( X2 != zero_zero_nat )
       => ( ( ( X2 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
              = ( vEBT_Leaf @ A2 @ $true ) ) )
          & ( ( X2 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
              = ( vEBT_Leaf @ A2 @ B3 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_5844_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_5845_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_5846_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_5847_nth__rotate1,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
        = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_5848_nth__rotate1,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
        = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_5849_nth__rotate1,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
        = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_5850_nth__rotate1,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_5851_ln__2__less__1,axiom,
    ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).

% ln_2_less_1
thf(fact_5852_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_5853_verit__le__mono__div,axiom,
    ! [A3: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B2 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_5854_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_5855_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A2 @ C ) ) @ ( times_times_real @ B3 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_5856_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_5857_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_5858_powr__neg__numeral,axiom,
    ! [X2: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_5859_pos__zdiv__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( divide_divide_int @ B3 @ A2 ) ) ) ).

% pos_zdiv_mult_2
thf(fact_5860_neg__zdiv__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( divide_divide_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A2 ) ) ) ).

% neg_zdiv_mult_2
thf(fact_5861_pos__zmod__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ A2 ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_5862_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_5863_real__exp__bound__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_5864_neg__zmod__mult__2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_5865_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A2: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B3 )
     => ( ( eucl_rel_int @ A2 @ B3 @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5866_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_5867_exp__lower__Taylor__quadratic,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_5868_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A2: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ B3 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B3 @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5869_arctan__double,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_5870_ln__one__minus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_5871_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( 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 @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_5872_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_5873_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5874_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X2 @ Xa2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( Y3
             != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
             => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y3
                      = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
                  & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y3
                      = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
thf(fact_5875_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X2: nat,Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X2 )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) )
      & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X2 )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).

% VEBT_internal.insert'.simps(2)
thf(fact_5876_insert__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
          = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ) ) ).

% insert_correct
thf(fact_5877_insert__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
          = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ) ) ).

% insert_corr
thf(fact_5878_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X2: code_integer] :
      ( ! [X5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
         => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5879_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X2: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5880_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X2: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
         => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5881_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X2: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5882_pred__list__to__short,axiom,
    ! [Deg: nat,X2: 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 @ X2 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_5883_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X2: 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 @ X2 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_5884_UnCI,axiom,
    ! [C: real,B2: set_real,A3: set_real] :
      ( ( ~ ( member_real @ C @ B2 )
       => ( member_real @ C @ A3 ) )
     => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5885_UnCI,axiom,
    ! [C: $o,B2: set_o,A3: set_o] :
      ( ( ~ ( member_o @ C @ B2 )
       => ( member_o @ C @ A3 ) )
     => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5886_UnCI,axiom,
    ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( ( ~ ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ A3 ) )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5887_UnCI,axiom,
    ! [C: int,B2: set_int,A3: set_int] :
      ( ( ~ ( member_int @ C @ B2 )
       => ( member_int @ C @ A3 ) )
     => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5888_UnCI,axiom,
    ! [C: nat,B2: set_nat,A3: set_nat] :
      ( ( ~ ( member_nat @ C @ B2 )
       => ( member_nat @ C @ A3 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5889_UnCI,axiom,
    ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( member8206827879206165904at_nat @ C @ B2 )
       => ( member8206827879206165904at_nat @ C @ A3 ) )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5890_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ C @ B2 )
       => ( member8757157785044589968at_nat @ C @ A3 ) )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).

% UnCI
thf(fact_5891_Un__iff,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) )
      = ( ( member_real @ C @ A3 )
        | ( member_real @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5892_Un__iff,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) )
      = ( ( member_o @ C @ A3 )
        | ( member_o @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5893_Un__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
      = ( ( member_set_nat @ C @ A3 )
        | ( member_set_nat @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5894_Un__iff,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) )
      = ( ( member_int @ C @ A3 )
        | ( member_int @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5895_Un__iff,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
      = ( ( member_nat @ C @ A3 )
        | ( member_nat @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5896_Un__iff,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
      = ( ( member8206827879206165904at_nat @ C @ A3 )
        | ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5897_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
      = ( ( member8757157785044589968at_nat @ C @ A3 )
        | ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_5898_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_5899_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% high_def
thf(fact_5900_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_5901_high__inv,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
        = Y3 ) ) ).

% high_inv
thf(fact_5902_Un__empty,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
        = bot_bo5327735625951526323at_nat )
      = ( ( A3 = bot_bo5327735625951526323at_nat )
        & ( B2 = bot_bo5327735625951526323at_nat ) ) ) ).

% Un_empty
thf(fact_5903_Un__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
        = bot_bo228742789529271731at_nat )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_5904_Un__empty,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ( sup_sup_set_real @ A3 @ B2 )
        = bot_bot_set_real )
      = ( ( A3 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% Un_empty
thf(fact_5905_Un__empty,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ( sup_sup_set_o @ A3 @ B2 )
        = bot_bot_set_o )
      = ( ( A3 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_5906_Un__empty,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B2 )
        = bot_bot_set_nat )
      = ( ( A3 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_5907_Un__empty,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ( sup_sup_set_int @ A3 @ B2 )
        = bot_bot_set_int )
      = ( ( A3 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_5908_finite__Un,axiom,
    ! [F2: set_int,G2: set_int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) )
      = ( ( finite_finite_int @ F2 )
        & ( finite_finite_int @ G2 ) ) ) ).

% finite_Un
thf(fact_5909_finite__Un,axiom,
    ! [F2: set_complex,G2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) )
      = ( ( finite3207457112153483333omplex @ F2 )
        & ( finite3207457112153483333omplex @ G2 ) ) ) ).

% finite_Un
thf(fact_5910_finite__Un,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G2 ) )
      = ( ( finite6177210948735845034at_nat @ F2 )
        & ( finite6177210948735845034at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_5911_finite__Un,axiom,
    ! [F2: set_Extended_enat,G2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ F2 @ G2 ) )
      = ( ( finite4001608067531595151d_enat @ F2 )
        & ( finite4001608067531595151d_enat @ G2 ) ) ) ).

% finite_Un
thf(fact_5912_finite__Un,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_5913_finite__Un,axiom,
    ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F2 @ G2 ) )
      = ( ( finite4392333629123659920at_nat @ F2 )
        & ( finite4392333629123659920at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_5914_finite__Un,axiom,
    ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F2 @ G2 ) )
      = ( ( finite4343798906461161616at_nat @ F2 )
        & ( finite4343798906461161616at_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_5915_Un__subset__iff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A3 @ C2 )
        & ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_5916_Un__subset__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
      = ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
        & ( ord_le3000389064537975527at_nat @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_5917_Un__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
      = ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
        & ( ord_le1268244103169919719at_nat @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_5918_Un__subset__iff,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ C2 )
      = ( ( ord_less_eq_set_int @ A3 @ C2 )
        & ( ord_less_eq_set_int @ B2 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_5919_Un__insert__left,axiom,
    ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
      = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5920_Un__insert__left,axiom,
    ! [A2: real,B2: set_real,C2: set_real] :
      ( ( sup_sup_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
      = ( insert_real @ A2 @ ( sup_sup_set_real @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5921_Un__insert__left,axiom,
    ! [A2: $o,B2: set_o,C2: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
      = ( insert_o @ A2 @ ( sup_sup_set_o @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5922_Un__insert__left,axiom,
    ! [A2: int,B2: set_int,C2: set_int] :
      ( ( sup_sup_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
      = ( insert_int @ A2 @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5923_Un__insert__left,axiom,
    ! [A2: nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
      = ( insert_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5924_Un__insert__left,axiom,
    ! [A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A2 @ B2 ) @ C2 )
      = ( insert5050368324300391991at_nat @ A2 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5925_Un__insert__left,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A2 @ B2 ) @ C2 )
      = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_5926_Un__insert__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
      = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5927_Un__insert__right,axiom,
    ! [A3: set_real,A2: real,B2: set_real] :
      ( ( sup_sup_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
      = ( insert_real @ A2 @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5928_Un__insert__right,axiom,
    ! [A3: set_o,A2: $o,B2: set_o] :
      ( ( sup_sup_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
      = ( insert_o @ A2 @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5929_Un__insert__right,axiom,
    ! [A3: set_int,A2: int,B2: set_int] :
      ( ( sup_sup_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
      = ( insert_int @ A2 @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5930_Un__insert__right,axiom,
    ! [A3: set_nat,A2: nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
      = ( insert_nat @ A2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5931_Un__insert__right,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ A2 @ B2 ) )
      = ( insert5050368324300391991at_nat @ A2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5932_Un__insert__right,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( insert9069300056098147895at_nat @ A2 @ B2 ) )
      = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_5933_Un__Diff__cancel,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) )
      = ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).

% Un_Diff_cancel
thf(fact_5934_Un__Diff__cancel,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) )
      = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).

% Un_Diff_cancel
thf(fact_5935_Un__Diff__cancel,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
      = ( sup_sup_set_nat @ A3 @ B2 ) ) ).

% Un_Diff_cancel
thf(fact_5936_Un__Diff__cancel2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) @ A3 )
      = ( sup_su718114333110466843at_nat @ B2 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_5937_Un__Diff__cancel2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) @ A3 )
      = ( sup_su5525570899277871387at_nat @ B2 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_5938_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A3 ) @ A3 )
      = ( sup_sup_set_nat @ B2 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_5939_Compl__Diff__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ B2 ) ) ).

% Compl_Diff_eq
thf(fact_5940_Compl__Diff__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ B2 ) ) ).

% Compl_Diff_eq
thf(fact_5941_Compl__Diff__eq,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ B2 ) ) ).

% Compl_Diff_eq
thf(fact_5942_case4_I11_J,axiom,
    ( ( mi != ma )
   => ! [I3: nat] :
        ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I3 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X4: nat] :
              ( ( ( ( vEBT_VEBT_high @ X4 @ na )
                  = I3 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X4 )
                & ( ord_less_eq_nat @ X4 @ ma ) ) ) ) ) ) ).

% case4(11)
thf(fact_5943_UnE,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) )
     => ( ~ ( member_real @ C @ A3 )
       => ( member_real @ C @ B2 ) ) ) ).

% UnE
thf(fact_5944_UnE,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) )
     => ( ~ ( member_o @ C @ A3 )
       => ( member_o @ C @ B2 ) ) ) ).

% UnE
thf(fact_5945_UnE,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
     => ( ~ ( member_set_nat @ C @ A3 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% UnE
thf(fact_5946_UnE,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) )
     => ( ~ ( member_int @ C @ A3 )
       => ( member_int @ C @ B2 ) ) ) ).

% UnE
thf(fact_5947_UnE,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
     => ( ~ ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B2 ) ) ) ).

% UnE
thf(fact_5948_UnE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
     => ( ~ ( member8206827879206165904at_nat @ C @ A3 )
       => ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).

% UnE
thf(fact_5949_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
     => ( ~ ( member8757157785044589968at_nat @ C @ A3 )
       => ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).

% UnE
thf(fact_5950_UnI1,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ A3 )
     => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5951_UnI1,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ A3 )
     => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5952_UnI1,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A3 )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5953_UnI1,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ A3 )
     => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5954_UnI1,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5955_UnI1,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ A3 )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5956_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ A3 )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).

% UnI1
thf(fact_5957_UnI2,axiom,
    ! [C: real,B2: set_real,A3: set_real] :
      ( ( member_real @ C @ B2 )
     => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5958_UnI2,axiom,
    ! [C: $o,B2: set_o,A3: set_o] :
      ( ( member_o @ C @ B2 )
     => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5959_UnI2,axiom,
    ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ C @ B2 )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5960_UnI2,axiom,
    ! [C: int,B2: set_int,A3: set_int] :
      ( ( member_int @ C @ B2 )
     => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5961_UnI2,axiom,
    ! [C: nat,B2: set_nat,A3: set_nat] :
      ( ( member_nat @ C @ B2 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5962_UnI2,axiom,
    ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ C @ B2 )
     => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5963_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ B2 )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).

% UnI2
thf(fact_5964_bex__Un,axiom,
    ! [A3: set_nat,B2: set_nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) )
            & ( P @ X ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( P @ X ) )
        | ? [X: nat] :
            ( ( member_nat @ X @ B2 )
            & ( P @ X ) ) ) ) ).

% bex_Un
thf(fact_5965_bex__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ? [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
            & ( P @ X ) ) )
      = ( ? [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A3 )
            & ( P @ X ) )
        | ? [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ B2 )
            & ( P @ X ) ) ) ) ).

% bex_Un
thf(fact_5966_bex__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ? [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
            & ( P @ X ) ) )
      = ( ? [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ A3 )
            & ( P @ X ) )
        | ? [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ B2 )
            & ( P @ X ) ) ) ) ).

% bex_Un
thf(fact_5967_ball__Un,axiom,
    ! [A3: set_nat,B2: set_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) )
           => ( P @ X ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( P @ X ) )
        & ! [X: nat] :
            ( ( member_nat @ X @ B2 )
           => ( P @ X ) ) ) ) ).

% ball_Un
thf(fact_5968_ball__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
           => ( P @ X ) ) )
      = ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A3 )
           => ( P @ X ) )
        & ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ B2 )
           => ( P @ X ) ) ) ) ).

% ball_Un
thf(fact_5969_ball__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ! [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
           => ( P @ X ) ) )
      = ( ! [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ A3 )
           => ( P @ X ) )
        & ! [X: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X @ B2 )
           => ( P @ X ) ) ) ) ).

% ball_Un
thf(fact_5970_Un__assoc,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
      = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% Un_assoc
thf(fact_5971_Un__assoc,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
      = ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).

% Un_assoc
thf(fact_5972_Un__assoc,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).

% Un_assoc
thf(fact_5973_Un__absorb,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_5974_Un__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_5975_Un__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_5976_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_5977_Un__commute,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_5978_Un__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B6 @ A6 ) ) ) ).

% Un_commute
thf(fact_5979_Un__left__absorb,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) )
      = ( sup_sup_set_nat @ A3 @ B2 ) ) ).

% Un_left_absorb
thf(fact_5980_Un__left__absorb,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
      = ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).

% Un_left_absorb
thf(fact_5981_Un__left__absorb,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
      = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).

% Un_left_absorb
thf(fact_5982_Un__left__commute,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
      = ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_5983_Un__left__commute,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
      = ( sup_su718114333110466843at_nat @ B2 @ ( sup_su718114333110466843at_nat @ A3 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_5984_Un__left__commute,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_5985_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X2 @ bot_bo5327735625951526323at_nat )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_5986_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X2 @ bot_bo228742789529271731at_nat )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_5987_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_real] :
      ( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_5988_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_5989_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_5990_boolean__algebra_Odisj__zero__right,axiom,
    ! [X2: set_int] :
      ( ( sup_sup_set_int @ X2 @ bot_bot_set_int )
      = X2 ) ).

% boolean_algebra.disj_zero_right
thf(fact_5991_Un__empty__right,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ bot_bo5327735625951526323at_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_5992_Un__empty__right,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ bot_bo228742789529271731at_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_5993_Un__empty__right,axiom,
    ! [A3: set_real] :
      ( ( sup_sup_set_real @ A3 @ bot_bot_set_real )
      = A3 ) ).

% Un_empty_right
thf(fact_5994_Un__empty__right,axiom,
    ! [A3: set_o] :
      ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Un_empty_right
thf(fact_5995_Un__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_5996_Un__empty__right,axiom,
    ! [A3: set_int] :
      ( ( sup_sup_set_int @ A3 @ bot_bot_set_int )
      = A3 ) ).

% Un_empty_right
thf(fact_5997_Un__empty__left,axiom,
    ! [B2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_5998_Un__empty__left,axiom,
    ! [B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_5999_Un__empty__left,axiom,
    ! [B2: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_6000_Un__empty__left,axiom,
    ! [B2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_6001_Un__empty__left,axiom,
    ! [B2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_6002_Un__empty__left,axiom,
    ! [B2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_6003_infinite__Un,axiom,
    ! [S: set_int,T2: set_int] :
      ( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) )
      = ( ~ ( finite_finite_int @ S )
        | ~ ( finite_finite_int @ T2 ) ) ) ).

% infinite_Un
thf(fact_6004_infinite__Un,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ( ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T2 ) ) )
      = ( ~ ( finite3207457112153483333omplex @ S )
        | ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).

% infinite_Un
thf(fact_6005_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) ) )
      = ( ~ ( finite6177210948735845034at_nat @ S )
        | ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_6006_infinite__Un,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ( ~ ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ S @ T2 ) ) )
      = ( ~ ( finite4001608067531595151d_enat @ S )
        | ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).

% infinite_Un
thf(fact_6007_infinite__Un,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_6008_infinite__Un,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) ) )
      = ( ~ ( finite4392333629123659920at_nat @ S )
        | ~ ( finite4392333629123659920at_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_6009_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) ) )
      = ( ~ ( finite4343798906461161616at_nat @ S )
        | ~ ( finite4343798906461161616at_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_6010_Un__infinite,axiom,
    ! [S: set_int,T2: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6011_Un__infinite,axiom,
    ! [S: set_complex,T2: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S )
     => ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6012_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6013_Un__infinite,axiom,
    ! [S: set_Extended_enat,T2: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S )
     => ~ ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6014_Un__infinite,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6015_Un__infinite,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ~ ( finite4392333629123659920at_nat @ S )
     => ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6016_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_6017_finite__UnI,axiom,
    ! [F2: set_int,G2: set_int] :
      ( ( finite_finite_int @ F2 )
     => ( ( finite_finite_int @ G2 )
       => ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6018_finite__UnI,axiom,
    ! [F2: set_complex,G2: set_complex] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( finite3207457112153483333omplex @ G2 )
       => ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6019_finite__UnI,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( finite6177210948735845034at_nat @ G2 )
       => ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6020_finite__UnI,axiom,
    ! [F2: set_Extended_enat,G2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( finite4001608067531595151d_enat @ G2 )
       => ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6021_finite__UnI,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( finite_finite_nat @ G2 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6022_finite__UnI,axiom,
    ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ F2 )
     => ( ( finite4392333629123659920at_nat @ G2 )
       => ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6023_finite__UnI,axiom,
    ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ F2 )
     => ( ( finite4343798906461161616at_nat @ G2 )
       => ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_6024_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_6025_subset__Un__eq,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_6026_subset__Un__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_6027_subset__Un__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( sup_sup_set_int @ A6 @ B6 )
          = B6 ) ) ) ).

% subset_Un_eq
thf(fact_6028_subset__UnE,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B2 ) )
     => ~ ! [A8: set_nat] :
            ( ( ord_less_eq_set_nat @ A8 @ A3 )
           => ! [B10: set_nat] :
                ( ( ord_less_eq_set_nat @ B10 @ B2 )
               => ( C2
                 != ( sup_sup_set_nat @ A8 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_6029_subset__UnE,axiom,
    ! [C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
     => ~ ! [A8: set_Pr8693737435421807431at_nat] :
            ( ( ord_le3000389064537975527at_nat @ A8 @ A3 )
           => ! [B10: set_Pr8693737435421807431at_nat] :
                ( ( ord_le3000389064537975527at_nat @ B10 @ B2 )
               => ( C2
                 != ( sup_su718114333110466843at_nat @ A8 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_6030_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
     => ~ ! [A8: set_Pr4329608150637261639at_nat] :
            ( ( ord_le1268244103169919719at_nat @ A8 @ A3 )
           => ! [B10: set_Pr4329608150637261639at_nat] :
                ( ( ord_le1268244103169919719at_nat @ B10 @ B2 )
               => ( C2
                 != ( sup_su5525570899277871387at_nat @ A8 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_6031_subset__UnE,axiom,
    ! [C2: set_int,A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A3 @ B2 ) )
     => ~ ! [A8: set_int] :
            ( ( ord_less_eq_set_int @ A8 @ A3 )
           => ! [B10: set_int] :
                ( ( ord_less_eq_set_int @ B10 @ B2 )
               => ( C2
                 != ( sup_sup_set_int @ A8 @ B10 ) ) ) ) ) ).

% subset_UnE
thf(fact_6032_Un__absorb2,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A3 )
     => ( ( sup_sup_set_nat @ A3 @ B2 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_6033_Un__absorb2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B2 @ A3 )
     => ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_6034_Un__absorb2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B2 @ A3 )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_6035_Un__absorb2,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A3 )
     => ( ( sup_sup_set_int @ A3 @ B2 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_6036_Un__absorb1,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( sup_sup_set_nat @ A3 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_6037_Un__absorb1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ B2 )
     => ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_6038_Un__absorb1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ B2 )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_6039_Un__absorb1,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( sup_sup_set_int @ A3 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_6040_Un__upper2,axiom,
    ! [B2: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).

% Un_upper2
thf(fact_6041_Un__upper2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).

% Un_upper2
thf(fact_6042_Un__upper2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).

% Un_upper2
thf(fact_6043_Un__upper2,axiom,
    ! [B2: set_int,A3: set_int] : ( ord_less_eq_set_int @ B2 @ ( sup_sup_set_int @ A3 @ B2 ) ) ).

% Un_upper2
thf(fact_6044_Un__upper1,axiom,
    ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).

% Un_upper1
thf(fact_6045_Un__upper1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).

% Un_upper1
thf(fact_6046_Un__upper1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).

% Un_upper1
thf(fact_6047_Un__upper1,axiom,
    ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ A3 @ B2 ) ) ).

% Un_upper1
thf(fact_6048_Un__least,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_6049_Un__least,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
     => ( ( ord_le3000389064537975527at_nat @ B2 @ C2 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_6050_Un__least,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ C2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_6051_Un__least,axiom,
    ! [A3: set_int,C2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ B2 @ C2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ C2 ) ) ) ).

% Un_least
thf(fact_6052_Un__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ D4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_6053_Un__mono,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,D4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
     => ( ( ord_le3000389064537975527at_nat @ B2 @ D4 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_6054_Un__mono,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B2 @ D4 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_6055_Un__mono,axiom,
    ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ B2 @ D4 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ ( sup_sup_set_int @ C2 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_6056_Un__Diff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
      = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) @ ( minus_8321449233255521966at_nat @ B2 @ C2 ) ) ) ).

% Un_Diff
thf(fact_6057_Un__Diff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) @ ( minus_3314409938677909166at_nat @ B2 @ C2 ) ) ) ).

% Un_Diff
thf(fact_6058_Un__Diff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).

% Un_Diff
thf(fact_6059_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M )
     => ( ( ord_less_eq_rat @ M @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
          = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_6060_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M: num,U: num] :
      ( ( ord_less_eq_num @ L @ M )
     => ( ( ord_less_eq_num @ M @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M ) @ ( set_or7049704709247886629st_num @ M @ U ) )
          = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_6061_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M )
     => ( ( ord_less_eq_nat @ M @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
          = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_6062_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M: int,U: int] :
      ( ( ord_less_eq_int @ L @ M )
     => ( ( ord_less_eq_int @ M @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
          = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_6063_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,M: real,U: real] :
      ( ( ord_less_eq_real @ L @ M )
     => ( ( ord_less_eq_real @ M @ U )
       => ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M ) @ ( set_or1222579329274155063t_real @ M @ U ) )
          = ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_6064_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A4: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_6065_insert__is__Un,axiom,
    ( insert5050368324300391991at_nat
    = ( ^ [A4: produc859450856879609959at_nat] : ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_6066_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A4: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_6067_insert__is__Un,axiom,
    ( insert_real
    = ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).

% insert_is_Un
thf(fact_6068_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A4: $o] : ( sup_sup_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_6069_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_6070_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A4: int] : ( sup_sup_set_int @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_6071_Un__singleton__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A3 @ B2 )
        = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A3 = bot_bo2099793752762293965at_nat )
          & ( B2
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A3
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
          & ( B2 = bot_bo2099793752762293965at_nat ) )
        | ( ( A3
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
          & ( B2
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6072_Un__singleton__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,X2: produc859450856879609959at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
        = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
      = ( ( ( A3 = bot_bo5327735625951526323at_nat )
          & ( B2
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) )
        | ( ( A3
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
          & ( B2 = bot_bo5327735625951526323at_nat ) )
        | ( ( A3
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
          & ( B2
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6073_Un__singleton__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X2: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
        = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A3 = bot_bo228742789529271731at_nat )
          & ( B2
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A3
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
          & ( B2 = bot_bo228742789529271731at_nat ) )
        | ( ( A3
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
          & ( B2
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6074_Un__singleton__iff,axiom,
    ! [A3: set_real,B2: set_real,X2: real] :
      ( ( ( sup_sup_set_real @ A3 @ B2 )
        = ( insert_real @ X2 @ bot_bot_set_real ) )
      = ( ( ( A3 = bot_bot_set_real )
          & ( B2
            = ( insert_real @ X2 @ bot_bot_set_real ) ) )
        | ( ( A3
            = ( insert_real @ X2 @ bot_bot_set_real ) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A3
            = ( insert_real @ X2 @ bot_bot_set_real ) )
          & ( B2
            = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6075_Un__singleton__iff,axiom,
    ! [A3: set_o,B2: set_o,X2: $o] :
      ( ( ( sup_sup_set_o @ A3 @ B2 )
        = ( insert_o @ X2 @ bot_bot_set_o ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B2
            = ( insert_o @ X2 @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B2
            = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6076_Un__singleton__iff,axiom,
    ! [A3: set_nat,B2: set_nat,X2: nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B2 )
        = ( insert_nat @ X2 @ bot_bot_set_nat ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B2
            = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat @ X2 @ bot_bot_set_nat ) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat @ X2 @ bot_bot_set_nat ) )
          & ( B2
            = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6077_Un__singleton__iff,axiom,
    ! [A3: set_int,B2: set_int,X2: int] :
      ( ( ( sup_sup_set_int @ A3 @ B2 )
        = ( insert_int @ X2 @ bot_bot_set_int ) )
      = ( ( ( A3 = bot_bot_set_int )
          & ( B2
            = ( insert_int @ X2 @ bot_bot_set_int ) ) )
        | ( ( A3
            = ( insert_int @ X2 @ bot_bot_set_int ) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A3
            = ( insert_int @ X2 @ bot_bot_set_int ) )
          & ( B2
            = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_6078_singleton__Un__iff,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bo2099793752762293965at_nat )
          & ( B2
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A3
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
          & ( B2 = bot_bo2099793752762293965at_nat ) )
        | ( ( A3
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
          & ( B2
            = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6079_singleton__Un__iff,axiom,
    ! [X2: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat )
        = ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bo5327735625951526323at_nat )
          & ( B2
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) )
        | ( ( A3
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
          & ( B2 = bot_bo5327735625951526323at_nat ) )
        | ( ( A3
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
          & ( B2
            = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6080_singleton__Un__iff,axiom,
    ! [X2: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bo228742789529271731at_nat )
          & ( B2
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A3
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
          & ( B2 = bot_bo228742789529271731at_nat ) )
        | ( ( A3
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
          & ( B2
            = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6081_singleton__Un__iff,axiom,
    ! [X2: real,A3: set_real,B2: set_real] :
      ( ( ( insert_real @ X2 @ bot_bot_set_real )
        = ( sup_sup_set_real @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bot_set_real )
          & ( B2
            = ( insert_real @ X2 @ bot_bot_set_real ) ) )
        | ( ( A3
            = ( insert_real @ X2 @ bot_bot_set_real ) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A3
            = ( insert_real @ X2 @ bot_bot_set_real ) )
          & ( B2
            = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6082_singleton__Un__iff,axiom,
    ! [X2: $o,A3: set_o,B2: set_o] :
      ( ( ( insert_o @ X2 @ bot_bot_set_o )
        = ( sup_sup_set_o @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B2
            = ( insert_o @ X2 @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B2
            = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6083_singleton__Un__iff,axiom,
    ! [X2: nat,A3: set_nat,B2: set_nat] :
      ( ( ( insert_nat @ X2 @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B2
            = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat @ X2 @ bot_bot_set_nat ) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat @ X2 @ bot_bot_set_nat ) )
          & ( B2
            = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6084_singleton__Un__iff,axiom,
    ! [X2: int,A3: set_int,B2: set_int] :
      ( ( ( insert_int @ X2 @ bot_bot_set_int )
        = ( sup_sup_set_int @ A3 @ B2 ) )
      = ( ( ( A3 = bot_bot_set_int )
          & ( B2
            = ( insert_int @ X2 @ bot_bot_set_int ) ) )
        | ( ( A3
            = ( insert_int @ X2 @ bot_bot_set_int ) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A3
            = ( insert_int @ X2 @ bot_bot_set_int ) )
          & ( B2
            = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_6085_Diff__subset__conv,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ C2 )
      = ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_6086_Diff__subset__conv,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ C2 )
      = ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_6087_Diff__subset__conv,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ C2 )
      = ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_6088_Diff__subset__conv,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ C2 )
      = ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_6089_Diff__partition,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A3 @ B2 )
     => ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) )
        = B2 ) ) ).

% Diff_partition
thf(fact_6090_Diff__partition,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ B2 )
     => ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) )
        = B2 ) ) ).

% Diff_partition
thf(fact_6091_Diff__partition,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
        = B2 ) ) ).

% Diff_partition
thf(fact_6092_Diff__partition,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( sup_sup_set_int @ A3 @ ( minus_minus_set_int @ B2 @ A3 ) )
        = B2 ) ) ).

% Diff_partition
thf(fact_6093_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_6094_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_6095_card__Un__le,axiom,
    ! [A3: set_complex,B2: set_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ).

% card_Un_le
thf(fact_6096_card__Un__le,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ).

% card_Un_le
thf(fact_6097_card__Un__le,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ).

% card_Un_le
thf(fact_6098_card__Un__le,axiom,
    ! [A3: set_int,B2: set_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ).

% card_Un_le
thf(fact_6099_card__Un__le,axiom,
    ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ).

% card_Un_le
thf(fact_6100_card__Un__le,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) ) ) ).

% card_Un_le
thf(fact_6101_card__Un__le,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B2 ) ) ) ).

% card_Un_le
thf(fact_6102_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K2: int] :
            ( ( P @ X5 )
            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X8: int] : ( P @ X8 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_6103_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X4 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_6104_aset_I5_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X4 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_6105_aset_I4_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 != T )
             => ( ( plus_plus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_6106_aset_I3_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 = T )
             => ( ( plus_plus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_6107_bset_I7_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X4 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_6108_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X4 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_6109_bset_I4_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 != T )
             => ( ( minus_minus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_6110_bset_I3_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 = T )
             => ( ( minus_minus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_6111_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X4 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_6112_aset_I6_J,axiom,
    ! [D4: int,T: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_6113_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X4 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_6114_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_6115_cppi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( P @ X5 )
              = ( P4 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A3 )
                     => ( X5
                       != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K2: int] :
                ( ( P4 @ X5 )
                = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A3 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_6116_cpmi,axiom,
    ! [D4: int,P: int > $o,P4: int > $o,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( P @ X5 )
              = ( P4 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B2 )
                     => ( X5
                       != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K2: int] :
                ( ( P4 @ X5 )
                = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P4 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B2 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_6117_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_6118_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A2: $o,B3: $o,X2: nat] :
      ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
      = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 ) ) ).

% VEBT_internal.insert'.simps(1)
thf(fact_6119_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X2 ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ 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_6120_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_6121_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( 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 ) @ X2 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_6122_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( ( ord_less_nat @ X2 @ Ma )
            & ( ord_less_nat @ Mi @ X2 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_6123_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: 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 ) @ X2 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_6124_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).

% both_member_options_ding
thf(fact_6125_bit__split__inv,axiom,
    ! [X2: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
      = X2 ) ).

% bit_split_inv
thf(fact_6126_Int__iff,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
      = ( ( member_real @ C @ A3 )
        & ( member_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_6127_Int__iff,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
      = ( ( member_o @ C @ A3 )
        & ( member_o @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_6128_Int__iff,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
      = ( ( member_set_nat @ C @ A3 )
        & ( member_set_nat @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_6129_Int__iff,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
      = ( ( member_int @ C @ A3 )
        & ( member_int @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_6130_Int__iff,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
      = ( ( member_nat @ C @ A3 )
        & ( member_nat @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_6131_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
      = ( ( member8440522571783428010at_nat @ C @ A3 )
        & ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_6132_IntI,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ A3 )
     => ( ( member_real @ C @ B2 )
       => ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).

% IntI
thf(fact_6133_IntI,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ A3 )
     => ( ( member_o @ C @ B2 )
       => ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).

% IntI
thf(fact_6134_IntI,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A3 )
     => ( ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).

% IntI
thf(fact_6135_IntI,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ A3 )
     => ( ( member_int @ C @ B2 )
       => ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).

% IntI
thf(fact_6136_IntI,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( ( member_nat @ C @ B2 )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).

% IntI
thf(fact_6137_IntI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A3 )
     => ( ( member8440522571783428010at_nat @ C @ B2 )
       => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).

% IntI
thf(fact_6138_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% low_def
thf(fact_6139_low__inv,axiom,
    ! [X2: nat,N: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
        = X2 ) ) ).

% low_inv
thf(fact_6140_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X2 )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_6141_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_real] :
      ( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_zero_left
thf(fact_6142_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_6143_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_6144_boolean__algebra_Oconj__zero__left,axiom,
    ! [X2: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X2 )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
thf(fact_6145_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X2 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_6146_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_real] :
      ( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_zero_right
thf(fact_6147_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_6148_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_6149_boolean__algebra_Oconj__zero__right,axiom,
    ! [X2: set_int] :
      ( ( inf_inf_set_int @ X2 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
thf(fact_6150_finite__Int,axiom,
    ! [F2: set_int,G2: set_int] :
      ( ( ( finite_finite_int @ F2 )
        | ( finite_finite_int @ G2 ) )
     => ( finite_finite_int @ ( inf_inf_set_int @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_6151_finite__Int,axiom,
    ! [F2: set_complex,G2: set_complex] :
      ( ( ( finite3207457112153483333omplex @ F2 )
        | ( finite3207457112153483333omplex @ G2 ) )
     => ( finite3207457112153483333omplex @ ( inf_inf_set_complex @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_6152_finite__Int,axiom,
    ! [F2: set_Extended_enat,G2: set_Extended_enat] :
      ( ( ( finite4001608067531595151d_enat @ F2 )
        | ( finite4001608067531595151d_enat @ G2 ) )
     => ( finite4001608067531595151d_enat @ ( inf_in8357106775501769908d_enat @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_6153_finite__Int,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G2 ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_6154_finite__Int,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite6177210948735845034at_nat @ F2 )
        | ( finite6177210948735845034at_nat @ G2 ) )
     => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_6155_Int__subset__iff,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B2 ) )
      = ( ( ord_less_eq_set_nat @ C2 @ A3 )
        & ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_6156_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
      = ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
        & ( ord_le3146513528884898305at_nat @ C2 @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_6157_Int__subset__iff,axiom,
    ! [C2: set_int,A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ ( inf_inf_set_int @ A3 @ B2 ) )
      = ( ( ord_less_eq_set_int @ C2 @ A3 )
        & ( ord_less_eq_set_int @ C2 @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_6158_Int__insert__right__if1,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( member_real @ A2 @ A3 )
     => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
        = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6159_Int__insert__right__if1,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ A2 @ A3 )
     => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
        = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6160_Int__insert__right__if1,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ A2 @ A3 )
     => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
        = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6161_Int__insert__right__if1,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( member_int @ A2 @ A3 )
     => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
        = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6162_Int__insert__right__if1,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ A2 @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6163_Int__insert__right__if1,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
        = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_6164_Int__insert__right__if0,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ~ ( member_real @ A2 @ A3 )
     => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
        = ( inf_inf_set_real @ A3 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_6165_Int__insert__right__if0,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ~ ( member_o @ A2 @ A3 )
     => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
        = ( inf_inf_set_o @ A3 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_6166_Int__insert__right__if0,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A2 @ A3 )
     => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
        = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_6167_Int__insert__right__if0,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ~ ( member_int @ A2 @ A3 )
     => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
        = ( inf_inf_set_int @ A3 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_6168_Int__insert__right__if0,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ A2 @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
        = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_6169_Int__insert__right__if0,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
        = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_6170_insert__inter__insert,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ ( insert_real @ A2 @ B2 ) )
      = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_6171_insert__inter__insert,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ ( insert_o @ A2 @ B2 ) )
      = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_6172_insert__inter__insert,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ ( insert_int @ A2 @ B2 ) )
      = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_6173_insert__inter__insert,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ ( insert_nat @ A2 @ B2 ) )
      = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_6174_insert__inter__insert,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
      = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_6175_Int__insert__left__if1,axiom,
    ! [A2: real,C2: set_real,B2: set_real] :
      ( ( member_real @ A2 @ C2 )
     => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
        = ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6176_Int__insert__left__if1,axiom,
    ! [A2: $o,C2: set_o,B2: set_o] :
      ( ( member_o @ A2 @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
        = ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6177_Int__insert__left__if1,axiom,
    ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ A2 @ C2 )
     => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
        = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6178_Int__insert__left__if1,axiom,
    ! [A2: int,C2: set_int,B2: set_int] :
      ( ( member_int @ A2 @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
        = ( insert_int @ A2 @ ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6179_Int__insert__left__if1,axiom,
    ! [A2: nat,C2: set_nat,B2: set_nat] :
      ( ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6180_Int__insert__left__if1,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A2 @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
        = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_6181_Int__insert__left__if0,axiom,
    ! [A2: real,C2: set_real,B2: set_real] :
      ( ~ ( member_real @ A2 @ C2 )
     => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
        = ( inf_inf_set_real @ B2 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_6182_Int__insert__left__if0,axiom,
    ! [A2: $o,C2: set_o,B2: set_o] :
      ( ~ ( member_o @ A2 @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
        = ( inf_inf_set_o @ B2 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_6183_Int__insert__left__if0,axiom,
    ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A2 @ C2 )
     => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
        = ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_6184_Int__insert__left__if0,axiom,
    ! [A2: int,C2: set_int,B2: set_int] :
      ( ~ ( member_int @ A2 @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
        = ( inf_inf_set_int @ B2 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_6185_Int__insert__left__if0,axiom,
    ! [A2: nat,C2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
        = ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_6186_Int__insert__left__if0,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_6187_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_6188_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_6189_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_6190_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_6191_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_6192_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_6193_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_6194_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Un_Int_eq(2)
thf(fact_6195_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_6196_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_6197_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ S @ ( sup_su718114333110466843at_nat @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_6198_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T2 ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_6199_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T2 @ ( sup_su6327502436637775413at_nat @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_6200_Un__Int__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_6201_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ T2 @ ( sup_su718114333110466843at_nat @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_6202_Un__Int__eq_I4_J,axiom,
    ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ T2 @ ( sup_su5525570899277871387at_nat @ S @ T2 ) )
      = T2 ) ).

% Un_Int_eq(4)
thf(fact_6203_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_6204_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_6205_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_6206_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T2 ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_6207_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_6208_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_6209_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_6210_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T2 ) @ T2 )
      = T2 ) ).

% Int_Un_eq(2)
thf(fact_6211_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_6212_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_6213_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ S @ ( inf_in4302113700860409141at_nat @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_6214_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T2 ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_6215_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T2 @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_6216_Int__Un__eq_I4_J,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_6217_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ T2 @ ( inf_in4302113700860409141at_nat @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_6218_Int__Un__eq_I4_J,axiom,
    ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ T2 @ ( inf_in7913087082777306421at_nat @ S @ T2 ) )
      = T2 ) ).

% Int_Un_eq(4)
thf(fact_6219_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_6220_inf__compl__bot__left1,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left1
thf(fact_6221_inf__compl__bot__left1,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ ( inf_inf_set_real @ X2 @ Y3 ) )
      = bot_bot_set_real ) ).

% inf_compl_bot_left1
thf(fact_6222_inf__compl__bot__left1,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ ( inf_inf_set_o @ X2 @ Y3 ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left1
thf(fact_6223_inf__compl__bot__left1,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( inf_inf_set_nat @ X2 @ Y3 ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left1
thf(fact_6224_inf__compl__bot__left1,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( inf_inf_set_int @ X2 @ Y3 ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left1
thf(fact_6225_inf__compl__bot__left2,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ Y3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left2
thf(fact_6226_inf__compl__bot__left2,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ Y3 ) )
      = bot_bot_set_real ) ).

% inf_compl_bot_left2
thf(fact_6227_inf__compl__bot__left2,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ Y3 ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left2
thf(fact_6228_inf__compl__bot__left2,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y3 ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left2
thf(fact_6229_inf__compl__bot__left2,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( inf_inf_set_int @ X2 @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left2
thf(fact_6230_inf__compl__bot__right,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ ( uminus6524753893492686040at_nat @ X2 ) ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_right
thf(fact_6231_inf__compl__bot__right,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ Y3 @ ( uminus612125837232591019t_real @ X2 ) ) )
      = bot_bot_set_real ) ).

% inf_compl_bot_right
thf(fact_6232_inf__compl__bot__right,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ Y3 @ ( uminus_uminus_set_o @ X2 ) ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_right
thf(fact_6233_inf__compl__bot__right,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X2 ) ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_right
thf(fact_6234_inf__compl__bot__right,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( inf_inf_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_right
thf(fact_6235_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ X2 )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6236_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X2: set_real] :
      ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ X2 )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6237_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ X2 )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6238_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6239_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X2: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ X2 )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6240_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ X2 ) )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6241_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X2: set_real] :
      ( ( inf_inf_set_real @ X2 @ ( uminus612125837232591019t_real @ X2 ) )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6242_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ X2 @ ( uminus_uminus_set_o @ X2 ) )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6243_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6244_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X2: set_int] :
      ( ( inf_inf_set_int @ X2 @ ( uminus1532241313380277803et_int @ X2 ) )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6245_disjoint__insert_I2_J,axiom,
    ! [A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ B3 @ B2 ) ) )
      = ( ~ ( member_set_nat @ B3 @ A3 )
        & ( bot_bot_set_set_nat
          = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6246_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B3 @ A3 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6247_disjoint__insert_I2_J,axiom,
    ! [A3: set_real,B3: real,B2: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ A3 @ ( insert_real @ B3 @ B2 ) ) )
      = ( ~ ( member_real @ B3 @ A3 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6248_disjoint__insert_I2_J,axiom,
    ! [A3: set_o,B3: $o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A3 @ ( insert_o @ B3 @ B2 ) ) )
      = ( ~ ( member_o @ B3 @ A3 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6249_disjoint__insert_I2_J,axiom,
    ! [A3: set_nat,B3: nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) )
      = ( ~ ( member_nat @ B3 @ A3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6250_disjoint__insert_I2_J,axiom,
    ! [A3: set_int,B3: int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A3 @ ( insert_int @ B3 @ B2 ) ) )
      = ( ~ ( member_int @ B3 @ A3 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_6251_disjoint__insert_I1_J,axiom,
    ! [B2: set_set_nat,A2: set_nat,A3: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A2 @ A3 ) )
        = bot_bot_set_set_nat )
      = ( ~ ( member_set_nat @ A2 @ B2 )
        & ( ( inf_inf_set_set_nat @ B2 @ A3 )
          = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6252_disjoint__insert_I1_J,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ A3 ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
        & ( ( inf_in2572325071724192079at_nat @ B2 @ A3 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6253_disjoint__insert_I1_J,axiom,
    ! [B2: set_real,A2: real,A3: set_real] :
      ( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A2 @ A3 ) )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A2 @ B2 )
        & ( ( inf_inf_set_real @ B2 @ A3 )
          = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
thf(fact_6254_disjoint__insert_I1_J,axiom,
    ! [B2: set_o,A2: $o,A3: set_o] :
      ( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A2 @ A3 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B2 )
        & ( ( inf_inf_set_o @ B2 @ A3 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_6255_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A2: nat,A3: set_nat] :
      ( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A2 @ A3 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B2 )
        & ( ( inf_inf_set_nat @ B2 @ A3 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_6256_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A2: int,A3: set_int] :
      ( ( ( inf_inf_set_int @ B2 @ ( insert_int @ A2 @ A3 ) )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A2 @ B2 )
        & ( ( inf_inf_set_int @ B2 @ A3 )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_6257_insert__disjoint_I2_J,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A3 ) @ B2 ) )
      = ( ~ ( member_set_nat @ A2 @ B2 )
        & ( bot_bot_set_set_nat
          = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6258_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ B2 ) )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6259_insert__disjoint_I2_J,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ B2 ) )
      = ( ~ ( member_real @ A2 @ B2 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6260_insert__disjoint_I2_J,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ B2 ) )
      = ( ~ ( member_o @ A2 @ B2 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6261_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B2 ) )
      = ( ~ ( member_nat @ A2 @ B2 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6262_insert__disjoint_I2_J,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ B2 ) )
      = ( ~ ( member_int @ A2 @ B2 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_6263_insert__disjoint_I1_J,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A3 ) @ B2 )
        = bot_bot_set_set_nat )
      = ( ~ ( member_set_nat @ A2 @ B2 )
        & ( ( inf_inf_set_set_nat @ A3 @ B2 )
          = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6264_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
        & ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6265_insert__disjoint_I1_J,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ B2 )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A2 @ B2 )
        & ( ( inf_inf_set_real @ A3 @ B2 )
          = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
thf(fact_6266_insert__disjoint_I1_J,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ B2 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B2 )
        & ( ( inf_inf_set_o @ A3 @ B2 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_6267_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B2 )
        & ( ( inf_inf_set_nat @ A3 @ B2 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_6268_insert__disjoint_I1_J,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ B2 )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A2 @ B2 )
        & ( ( inf_inf_set_int @ A3 @ B2 )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_6269_Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_6270_Diff__disjoint,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real @ A3 @ ( minus_minus_set_real @ B2 @ A3 ) )
      = bot_bot_set_real ) ).

% Diff_disjoint
thf(fact_6271_Diff__disjoint,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ A3 @ ( minus_minus_set_o @ B2 @ A3 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_6272_Diff__disjoint,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ A3 @ ( minus_minus_set_int @ B2 @ A3 ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_6273_Diff__disjoint,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_6274_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_6275_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_6276_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_6277_Compl__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
thf(fact_6278_Compl__disjoint,axiom,
    ! [A3: set_real] :
      ( ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
      = bot_bot_set_real ) ).

% Compl_disjoint
thf(fact_6279_Compl__disjoint,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_6280_Compl__disjoint,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_6281_Compl__disjoint,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_6282_Compl__disjoint2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ A3 )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
thf(fact_6283_Compl__disjoint2,axiom,
    ! [A3: set_real] :
      ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ A3 ) @ A3 )
      = bot_bot_set_real ) ).

% Compl_disjoint2
thf(fact_6284_Compl__disjoint2,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A3 ) @ A3 )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_6285_Compl__disjoint2,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ A3 )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_6286_Compl__disjoint2,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ A3 )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_6287_Diff__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B2 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).

% Diff_Compl
thf(fact_6288_Diff__Compl,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B2 ) )
      = ( inf_inf_set_nat @ A3 @ B2 ) ) ).

% Diff_Compl
thf(fact_6289_Int__left__commute,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
      = ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_6290_Int__left__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ B2 @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_6291_Int__left__absorb,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B2 ) )
      = ( inf_inf_set_nat @ A3 @ B2 ) ) ).

% Int_left_absorb
thf(fact_6292_Int__left__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).

% Int_left_absorb
thf(fact_6293_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ B6 @ A6 ) ) ) ).

% Int_commute
thf(fact_6294_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B6 @ A6 ) ) ) ).

% Int_commute
thf(fact_6295_Int__absorb,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_6296_Int__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_6297_Int__assoc,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
      = ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).

% Int_assoc
thf(fact_6298_Int__assoc,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).

% Int_assoc
thf(fact_6299_IntD2,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
     => ( member_real @ C @ B2 ) ) ).

% IntD2
thf(fact_6300_IntD2,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
     => ( member_o @ C @ B2 ) ) ).

% IntD2
thf(fact_6301_IntD2,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
     => ( member_set_nat @ C @ B2 ) ) ).

% IntD2
thf(fact_6302_IntD2,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
     => ( member_int @ C @ B2 ) ) ).

% IntD2
thf(fact_6303_IntD2,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
     => ( member_nat @ C @ B2 ) ) ).

% IntD2
thf(fact_6304_IntD2,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
     => ( member8440522571783428010at_nat @ C @ B2 ) ) ).

% IntD2
thf(fact_6305_IntD1,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
     => ( member_real @ C @ A3 ) ) ).

% IntD1
thf(fact_6306_IntD1,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
     => ( member_o @ C @ A3 ) ) ).

% IntD1
thf(fact_6307_IntD1,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
     => ( member_set_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_6308_IntD1,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
     => ( member_int @ C @ A3 ) ) ).

% IntD1
thf(fact_6309_IntD1,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
     => ( member_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_6310_IntD1,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
     => ( member8440522571783428010at_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_6311_IntE,axiom,
    ! [C: real,A3: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
     => ~ ( ( member_real @ C @ A3 )
         => ~ ( member_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_6312_IntE,axiom,
    ! [C: $o,A3: set_o,B2: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
     => ~ ( ( member_o @ C @ A3 )
         => ~ ( member_o @ C @ B2 ) ) ) ).

% IntE
thf(fact_6313_IntE,axiom,
    ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
     => ~ ( ( member_set_nat @ C @ A3 )
         => ~ ( member_set_nat @ C @ B2 ) ) ) ).

% IntE
thf(fact_6314_IntE,axiom,
    ! [C: int,A3: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
     => ~ ( ( member_int @ C @ A3 )
         => ~ ( member_int @ C @ B2 ) ) ) ).

% IntE
thf(fact_6315_IntE,axiom,
    ! [C: nat,A3: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
     => ~ ( ( member_nat @ C @ A3 )
         => ~ ( member_nat @ C @ B2 ) ) ) ).

% IntE
thf(fact_6316_IntE,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A3 )
         => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% IntE
thf(fact_6317_Int__emptyI,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A3 )
         => ~ ( member_set_nat @ X5 @ B2 ) )
     => ( ( inf_inf_set_set_nat @ A3 @ B2 )
        = bot_bot_set_set_nat ) ) ).

% Int_emptyI
thf(fact_6318_Int__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A3 )
         => ~ ( member8440522571783428010at_nat @ X5 @ B2 ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_6319_Int__emptyI,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ~ ( member_real @ X5 @ B2 ) )
     => ( ( inf_inf_set_real @ A3 @ B2 )
        = bot_bot_set_real ) ) ).

% Int_emptyI
thf(fact_6320_Int__emptyI,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ~ ( member_o @ X5 @ B2 ) )
     => ( ( inf_inf_set_o @ A3 @ B2 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_6321_Int__emptyI,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ~ ( member_nat @ X5 @ B2 ) )
     => ( ( inf_inf_set_nat @ A3 @ B2 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_6322_Int__emptyI,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ~ ( member_int @ X5 @ B2 ) )
     => ( ( inf_inf_set_int @ A3 @ B2 )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_6323_disjoint__iff,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ A3 @ B2 )
        = bot_bot_set_set_nat )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
           => ~ ( member_set_nat @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_6324_disjoint__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A3 )
           => ~ ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_6325_disjoint__iff,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A3 @ B2 )
        = bot_bot_set_real )
      = ( ! [X: real] :
            ( ( member_real @ X @ A3 )
           => ~ ( member_real @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_6326_disjoint__iff,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B2 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A3 )
           => ~ ( member_o @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_6327_disjoint__iff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B2 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ~ ( member_nat @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_6328_disjoint__iff,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B2 )
        = bot_bot_set_int )
      = ( ! [X: int] :
            ( ( member_int @ X @ A3 )
           => ~ ( member_int @ X @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_6329_Int__empty__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B2 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_6330_Int__empty__left,axiom,
    ! [B2: set_real] :
      ( ( inf_inf_set_real @ bot_bot_set_real @ B2 )
      = bot_bot_set_real ) ).

% Int_empty_left
thf(fact_6331_Int__empty__left,axiom,
    ! [B2: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_6332_Int__empty__left,axiom,
    ! [B2: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_6333_Int__empty__left,axiom,
    ! [B2: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B2 )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_6334_Int__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_6335_Int__empty__right,axiom,
    ! [A3: set_real] :
      ( ( inf_inf_set_real @ A3 @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% Int_empty_right
thf(fact_6336_Int__empty__right,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ A3 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_6337_Int__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_6338_Int__empty__right,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ A3 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_6339_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A3 )
           => ! [Y: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y @ B2 )
               => ( X != Y ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_6340_disjoint__iff__not__equal,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A3 @ B2 )
        = bot_bot_set_real )
      = ( ! [X: real] :
            ( ( member_real @ X @ A3 )
           => ! [Y: real] :
                ( ( member_real @ Y @ B2 )
               => ( X != Y ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_6341_disjoint__iff__not__equal,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B2 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A3 )
           => ! [Y: $o] :
                ( ( member_o @ Y @ B2 )
               => ( X = ~ Y ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_6342_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B2 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ! [Y: nat] :
                ( ( member_nat @ Y @ B2 )
               => ( X != Y ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_6343_disjoint__iff__not__equal,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B2 )
        = bot_bot_set_int )
      = ( ! [X: int] :
            ( ( member_int @ X @ A3 )
           => ! [Y: int] :
                ( ( member_int @ Y @ B2 )
               => ( X != Y ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_6344_Int__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ B2 @ D4 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_6345_Int__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ D4 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ C2 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_6346_Int__mono,axiom,
    ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ C2 )
     => ( ( ord_less_eq_set_int @ B2 @ D4 )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ ( inf_inf_set_int @ C2 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_6347_Int__lower1,axiom,
    ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ A3 ) ).

% Int_lower1
thf(fact_6348_Int__lower1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ A3 ) ).

% Int_lower1
thf(fact_6349_Int__lower1,axiom,
    ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ A3 ) ).

% Int_lower1
thf(fact_6350_Int__lower2,axiom,
    ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_6351_Int__lower2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_6352_Int__lower2,axiom,
    ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_6353_Int__absorb1,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_6354_Int__absorb1,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_6355_Int__absorb1,axiom,
    ! [B2: set_int,A3: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A3 )
     => ( ( inf_inf_set_int @ A3 @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_6356_Int__absorb2,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( inf_inf_set_nat @ A3 @ B2 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_6357_Int__absorb2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_6358_Int__absorb2,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ( inf_inf_set_int @ A3 @ B2 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_6359_Int__greatest,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A3 )
     => ( ( ord_less_eq_set_nat @ C2 @ B2 )
       => ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_6360_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ C2 @ B2 )
       => ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_6361_Int__greatest,axiom,
    ! [C2: set_int,A3: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ A3 )
     => ( ( ord_less_eq_set_int @ C2 @ B2 )
       => ( ord_less_eq_set_int @ C2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_6362_Int__Collect__mono,axiom,
    ! [A3: set_o,B2: set_o,P: $o > $o,Q: $o > $o] :
      ( ( ord_less_eq_set_o @ A3 @ B2 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B2 @ ( collect_o @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6363_Int__Collect__mono,axiom,
    ! [A3: set_real,B2: set_real,P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ A3 @ B2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B2 @ ( collect_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6364_Int__Collect__mono,axiom,
    ! [A3: set_list_nat,B2: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
     => ( ! [X5: list_nat] :
            ( ( member_list_nat @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B2 @ ( collect_list_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6365_Int__Collect__mono,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
     => ( ! [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6366_Int__Collect__mono,axiom,
    ! [A3: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6367_Int__Collect__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
     => ( ! [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B2 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6368_Int__Collect__mono,axiom,
    ! [A3: set_int,B2: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A3 @ B2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B2 @ ( collect_int @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_6369_Int__insert__right,axiom,
    ! [A2: real,A3: set_real,B2: set_real] :
      ( ( ( member_real @ A2 @ A3 )
       => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
          = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) )
      & ( ~ ( member_real @ A2 @ A3 )
       => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
          = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_6370_Int__insert__right,axiom,
    ! [A2: $o,A3: set_o,B2: set_o] :
      ( ( ( member_o @ A2 @ A3 )
       => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
          = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) )
      & ( ~ ( member_o @ A2 @ A3 )
       => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
          = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_6371_Int__insert__right,axiom,
    ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
      ( ( ( member_set_nat @ A2 @ A3 )
       => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
          = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) )
      & ( ~ ( member_set_nat @ A2 @ A3 )
       => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
          = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_6372_Int__insert__right,axiom,
    ! [A2: int,A3: set_int,B2: set_int] :
      ( ( ( member_int @ A2 @ A3 )
       => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
          = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) )
      & ( ~ ( member_int @ A2 @ A3 )
       => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
          = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_6373_Int__insert__right,axiom,
    ! [A2: nat,A3: set_nat,B2: set_nat] :
      ( ( ( member_nat @ A2 @ A3 )
       => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
          = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) )
      & ( ~ ( member_nat @ A2 @ A3 )
       => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
          = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_6374_Int__insert__right,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A2 @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
          = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
          = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_6375_Int__insert__left,axiom,
    ! [A2: real,C2: set_real,B2: set_real] :
      ( ( ( member_real @ A2 @ C2 )
       => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
          = ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C2 ) ) ) )
      & ( ~ ( member_real @ A2 @ C2 )
       => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
          = ( inf_inf_set_real @ B2 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_6376_Int__insert__left,axiom,
    ! [A2: $o,C2: set_o,B2: set_o] :
      ( ( ( member_o @ A2 @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
          = ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C2 ) ) ) )
      & ( ~ ( member_o @ A2 @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
          = ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_6377_Int__insert__left,axiom,
    ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
      ( ( ( member_set_nat @ A2 @ C2 )
       => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
          = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) )
      & ( ~ ( member_set_nat @ A2 @ C2 )
       => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
          = ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_6378_Int__insert__left,axiom,
    ! [A2: int,C2: set_int,B2: set_int] :
      ( ( ( member_int @ A2 @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
          = ( insert_int @ A2 @ ( inf_inf_set_int @ B2 @ C2 ) ) ) )
      & ( ~ ( member_int @ A2 @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
          = ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_6379_Int__insert__left,axiom,
    ! [A2: nat,C2: set_nat,B2: set_nat] :
      ( ( ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
          = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
      & ( ~ ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
          = ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_6380_Int__insert__left,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A2 @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
          = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
          = ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_6381_Un__Int__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) @ A3 )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ A3 ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_6382_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A3 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A3 ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_6383_Un__Int__distrib2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) @ A3 )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B2 @ A3 ) @ ( sup_su718114333110466843at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_6384_Un__Int__distrib2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) @ A3 )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ A3 ) @ ( sup_su5525570899277871387at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_6385_Int__Un__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) @ A3 )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_6386_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A3 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_6387_Int__Un__distrib2,axiom,
    ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) @ A3 )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B2 @ A3 ) @ ( inf_in4302113700860409141at_nat @ C2 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_6388_Int__Un__distrib2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A3 )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ A3 ) @ ( inf_in7913087082777306421at_nat @ C2 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_6389_Un__Int__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) @ ( sup_su6327502436637775413at_nat @ A3 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_6390_Un__Int__distrib,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_6391_Un__Int__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) )
      = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ A3 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_6392_Un__Int__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_6393_Int__Un__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_6394_Int__Un__distrib,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_6395_Int__Un__distrib,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
      = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ A3 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_6396_Int__Un__distrib,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A3 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_6397_Un__Int__crazy,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_6398_Un__Int__crazy,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A3 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_6399_Un__Int__crazy,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) ) @ ( inf_in4302113700860409141at_nat @ C2 @ A3 ) )
      = ( inf_in4302113700860409141at_nat @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) @ ( sup_su718114333110466843at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_6400_Un__Int__crazy,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) @ ( inf_in7913087082777306421at_nat @ C2 @ A3 ) )
      = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) @ ( sup_su5525570899277871387at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_6401_Diff__Int__distrib2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ C2 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_6402_Diff__Int__distrib2,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ C2 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_6403_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ B2 ) ) ) ).

% Diff_Int_distrib
thf(fact_6404_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A3: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A3 @ B2 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).

% Diff_Int_distrib
thf(fact_6405_Diff__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).

% Diff_Diff_Int
thf(fact_6406_Diff__Diff__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B2 ) )
      = ( inf_inf_set_nat @ A3 @ B2 ) ) ).

% Diff_Diff_Int
thf(fact_6407_Diff__Int2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ B2 ) ) ).

% Diff_Int2
thf(fact_6408_Diff__Int2,axiom,
    ! [A3: set_nat,C2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ B2 ) ) ).

% Diff_Int2
thf(fact_6409_Int__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) ) ) ).

% Int_Diff
thf(fact_6410_Int__Diff,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
      = ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).

% Int_Diff
thf(fact_6411_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_6412_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_6413_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_6414_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_6415_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_6416_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_6417_inf__cancel__left1,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ A2 ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ B3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left1
thf(fact_6418_inf__cancel__left1,axiom,
    ! [X2: set_real,A2: set_real,B3: set_real] :
      ( ( inf_inf_set_real @ ( inf_inf_set_real @ X2 @ A2 ) @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ B3 ) )
      = bot_bot_set_real ) ).

% inf_cancel_left1
thf(fact_6419_inf__cancel__left1,axiom,
    ! [X2: set_o,A2: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ X2 @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ B3 ) )
      = bot_bot_set_o ) ).

% inf_cancel_left1
thf(fact_6420_inf__cancel__left1,axiom,
    ! [X2: set_nat,A2: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B3 ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left1
thf(fact_6421_inf__cancel__left1,axiom,
    ! [X2: set_int,A2: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ X2 @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ B3 ) )
      = bot_bot_set_int ) ).

% inf_cancel_left1
thf(fact_6422_inf__cancel__left2,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ A2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ B3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left2
thf(fact_6423_inf__cancel__left2,axiom,
    ! [X2: set_real,A2: set_real,B3: set_real] :
      ( ( inf_inf_set_real @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ A2 ) @ ( inf_inf_set_real @ X2 @ B3 ) )
      = bot_bot_set_real ) ).

% inf_cancel_left2
thf(fact_6424_inf__cancel__left2,axiom,
    ! [X2: set_o,A2: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ A2 ) @ ( inf_inf_set_o @ X2 @ B3 ) )
      = bot_bot_set_o ) ).

% inf_cancel_left2
thf(fact_6425_inf__cancel__left2,axiom,
    ! [X2: set_nat,A2: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A2 ) @ ( inf_inf_set_nat @ X2 @ B3 ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left2
thf(fact_6426_inf__cancel__left2,axiom,
    ! [X2: set_int,A2: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ A2 ) @ ( inf_inf_set_int @ X2 @ B3 ) )
      = bot_bot_set_int ) ).

% inf_cancel_left2
thf(fact_6427_Diff__triv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A3 @ B2 )
        = A3 ) ) ).

% Diff_triv
thf(fact_6428_Diff__triv,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A3 @ B2 )
        = bot_bot_set_real )
     => ( ( minus_minus_set_real @ A3 @ B2 )
        = A3 ) ) ).

% Diff_triv
thf(fact_6429_Diff__triv,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B2 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A3 @ B2 )
        = A3 ) ) ).

% Diff_triv
thf(fact_6430_Diff__triv,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B2 )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A3 @ B2 )
        = A3 ) ) ).

% Diff_triv
thf(fact_6431_Diff__triv,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B2 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A3 @ B2 )
        = A3 ) ) ).

% Diff_triv
thf(fact_6432_Int__Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_6433_Int__Diff__disjoint,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( inf_inf_set_real @ ( inf_inf_set_real @ A3 @ B2 ) @ ( minus_minus_set_real @ A3 @ B2 ) )
      = bot_bot_set_real ) ).

% Int_Diff_disjoint
thf(fact_6434_Int__Diff__disjoint,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A3 @ B2 ) @ ( minus_minus_set_o @ A3 @ B2 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_6435_Int__Diff__disjoint,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ ( minus_minus_set_int @ A3 @ B2 ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_6436_Int__Diff__disjoint,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ B2 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_6437_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) )
      = ( ord_le3146513528884898305at_nat @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_6438_Un__Int__assoc__eq,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
        = ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
      = ( ord_less_eq_set_nat @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_6439_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ C2 )
        = ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) )
      = ( ord_le3000389064537975527at_nat @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_6440_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ C2 )
        = ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) )
      = ( ord_le1268244103169919719at_nat @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_6441_Un__Int__assoc__eq,axiom,
    ! [A3: set_int,B2: set_int,C2: set_int] :
      ( ( ( sup_sup_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ C2 )
        = ( inf_inf_set_int @ A3 @ ( sup_sup_set_int @ B2 @ C2 ) ) )
      = ( ord_less_eq_set_int @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_6442_Un__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_6443_Un__Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_6444_Un__Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_6445_Un__Diff__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ A3 @ B2 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_6446_Int__Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_6447_Int__Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_6448_Int__Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_6449_Int__Diff__Un,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ B2 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_6450_Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).

% Diff_Int
thf(fact_6451_Diff__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) )
      = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) ) ) ).

% Diff_Int
thf(fact_6452_Diff__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) ) ) ).

% Diff_Int
thf(fact_6453_Diff__Int,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).

% Diff_Int
thf(fact_6454_Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).

% Diff_Un
thf(fact_6455_Diff__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
      ( ( minus_8321449233255521966at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
      = ( inf_in4302113700860409141at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) ) ) ).

% Diff_Un
thf(fact_6456_Diff__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
      = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) ) ) ).

% Diff_Un
thf(fact_6457_Diff__Un,axiom,
    ! [A3: set_nat,B2: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).

% Diff_Un
thf(fact_6458_Compl__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).

% Compl_Int
thf(fact_6459_Compl__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A3 @ B2 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).

% Compl_Int
thf(fact_6460_Compl__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B2 ) ) ) ).

% Compl_Int
thf(fact_6461_Compl__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).

% Compl_Int
thf(fact_6462_Compl__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).

% Compl_Un
thf(fact_6463_Compl__Un,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A3 @ B2 ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).

% Compl_Un
thf(fact_6464_Compl__Un,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( uminus4384627049435823934at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
      = ( inf_in4302113700860409141at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B2 ) ) ) ).

% Compl_Un
thf(fact_6465_Compl__Un,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
      = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).

% Compl_Un
thf(fact_6466_Diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ A6 @ ( uminus6524753893492686040at_nat @ B6 ) ) ) ) ).

% Diff_eq
thf(fact_6467_Diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ A6 @ ( uminus5710092332889474511et_nat @ B6 ) ) ) ) ).

% Diff_eq
thf(fact_6468_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_6469_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_6470_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_6471_inf__shunt,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y3 ) ) ) ).

% inf_shunt
thf(fact_6472_inf__shunt,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( ( inf_inf_set_real @ X2 @ Y3 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X2 @ ( uminus612125837232591019t_real @ Y3 ) ) ) ).

% inf_shunt
thf(fact_6473_inf__shunt,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( ( inf_inf_set_o @ X2 @ Y3 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X2 @ ( uminus_uminus_set_o @ Y3 ) ) ) ).

% inf_shunt
thf(fact_6474_inf__shunt,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ X2 @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).

% inf_shunt
thf(fact_6475_inf__shunt,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ( inf_inf_set_int @ X2 @ Y3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).

% inf_shunt
thf(fact_6476_shunt1,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Z )
      = ( ord_le3146513528884898305at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y3 ) @ Z ) ) ) ).

% shunt1
thf(fact_6477_shunt1,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z )
      = ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ Z ) ) ) ).

% shunt1
thf(fact_6478_shunt1,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ Y3 ) @ Z )
      = ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ Y3 ) @ Z ) ) ) ).

% shunt1
thf(fact_6479_shunt1,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y3 ) @ Z )
      = ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y3 ) @ Z ) ) ) ).

% shunt1
thf(fact_6480_shunt1,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Z )
      = ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ Z ) ) ) ).

% shunt1
thf(fact_6481_shunt2,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y3 ) ) @ Z )
      = ( ord_le3146513528884898305at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z ) ) ) ).

% shunt2
thf(fact_6482_shunt2,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) @ Z )
      = ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).

% shunt2
thf(fact_6483_shunt2,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ ( uminus4384627049435823934at_nat @ Y3 ) ) @ Z )
      = ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) ) ) ).

% shunt2
thf(fact_6484_shunt2,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ ( uminus935396558254630718at_nat @ Y3 ) ) @ Z )
      = ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) ) ) ).

% shunt2
thf(fact_6485_shunt2,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) @ Z )
      = ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ Y3 @ Z ) ) ) ).

% shunt2
thf(fact_6486_sup__neg__inf,axiom,
    ! [P6: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ P6 @ ( sup_su6327502436637775413at_nat @ Q3 @ R2 ) )
      = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P6 @ ( uminus6524753893492686040at_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6487_sup__neg__inf,axiom,
    ! [P6: set_nat,Q3: set_nat,R2: set_nat] :
      ( ( ord_less_eq_set_nat @ P6 @ ( sup_sup_set_nat @ Q3 @ R2 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P6 @ ( uminus5710092332889474511et_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6488_sup__neg__inf,axiom,
    ! [P6: set_Pr8693737435421807431at_nat,Q3: set_Pr8693737435421807431at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ P6 @ ( sup_su718114333110466843at_nat @ Q3 @ R2 ) )
      = ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ P6 @ ( uminus4384627049435823934at_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6489_sup__neg__inf,axiom,
    ! [P6: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ P6 @ ( sup_su5525570899277871387at_nat @ Q3 @ R2 ) )
      = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P6 @ ( uminus935396558254630718at_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6490_sup__neg__inf,axiom,
    ! [P6: set_int,Q3: set_int,R2: set_int] :
      ( ( ord_less_eq_set_int @ P6 @ ( sup_sup_set_int @ Q3 @ R2 ) )
      = ( ord_less_eq_set_int @ ( inf_inf_set_int @ P6 @ ( uminus1532241313380277803et_int @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6491_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6492_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A3 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6493_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6494_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6495_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6496_card__Un__Int,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
          = ( plus_plus_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6497_card__Un__Int,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
          = ( plus_plus_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6498_card__Un__Int,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
          = ( plus_plus_nat @ ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6499_card__Un__Int,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
          = ( plus_plus_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6500_card__Un__Int,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( plus_plus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
          = ( plus_plus_nat @ ( finite121521170596916366d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6501_card__Un__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
          = ( plus_plus_nat @ ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6502_card__Un__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
          = ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6503_card__Un__Int,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B2 )
       => ( ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) )
          = ( plus_plus_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) @ ( finite1207074278014112911at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6504_card__Un__Int,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A3 )
     => ( ( finite4343798906461161616at_nat @ B2 )
       => ( ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B2 ) )
          = ( plus_plus_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) @ ( finite3771342082235030671at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_6505_card__Diff__subset__Int,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) )
     => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6506_card__Diff__subset__Int,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
     => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6507_card__Diff__subset__Int,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( finite_finite_int @ ( inf_inf_set_int @ A3 @ B2 ) )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6508_card__Diff__subset__Int,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( inf_inf_set_complex @ A3 @ B2 ) )
     => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6509_card__Diff__subset__Int,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
     => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6510_card__Diff__subset__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6511_card__Diff__subset__Int,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ ( inf_inf_set_nat @ A3 @ B2 ) )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_6512_card__Un__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A3 @ B2 )
            = bot_bot_set_complex )
         => ( ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6513_card__Un__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
            = bot_bo7653980558646680370d_enat )
         => ( ( finite121521170596916366d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6514_card__Un__disjoint,axiom,
    ! [A3: set_real,B2: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A3 @ B2 )
            = bot_bot_set_real )
         => ( ( finite_card_real @ ( sup_sup_set_real @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6515_card__Un__disjoint,axiom,
    ! [A3: set_o,B2: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B2 )
       => ( ( ( inf_inf_set_o @ A3 @ B2 )
            = bot_bot_set_o )
         => ( ( finite_card_o @ ( sup_sup_set_o @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6516_card__Un__disjoint,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ( inf_inf_set_nat @ A3 @ B2 )
            = bot_bot_set_nat )
         => ( ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6517_card__Un__disjoint,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ( inf_inf_set_int @ A3 @ B2 )
            = bot_bot_set_int )
         => ( ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6518_card__Un__disjoint,axiom,
    ! [A3: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ( inf_inf_set_list_nat @ A3 @ B2 )
            = bot_bot_set_list_nat )
         => ( ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6519_card__Un__disjoint,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ( inf_inf_set_set_nat @ A3 @ B2 )
            = bot_bot_set_set_nat )
         => ( ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6520_card__Un__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
            = bot_bo2099793752762293965at_nat )
         => ( ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6521_card__Un__disjoint,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B2 )
       => ( ( ( inf_in4302113700860409141at_nat @ A3 @ B2 )
            = bot_bo5327735625951526323at_nat )
         => ( ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_6522_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_6523_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_6524_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_6525_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A4: $o,B4: $o] :
                ( A12
                = ( vEBT_Leaf @ A4 @ B4 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A23
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A23
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A23
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A23
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_6526_invar__vebt_Ocases,axiom,
    ! [A13: vEBT_VEBT,A24: nat] :
      ( ( vEBT_invar_vebt @ A13 @ A24 )
     => ( ( ? [A: $o,B: $o] :
              ( A13
              = ( vEBT_Leaf @ A @ B ) )
         => ( A24
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( A24 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ( vEBT_invar_vebt @ X4 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M4 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A13
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( A24 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ( vEBT_invar_vebt @ X4 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M4 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A13
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
                 => ( ( A24 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_invar_vebt @ X4 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M4 ) )
                             => ( ! [I3: nat] :
                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I3 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X4: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                                                        = I3 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X4 )
                                                      & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A13
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
                   => ( ( A24 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                           => ( vEBT_invar_vebt @ X4 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M4 ) )
                               => ( ! [I3: nat] :
                                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I3 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X4: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                                                          = I3 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X4 )
                                                        & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_6527_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N2: nat,TreeList4: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).

% in_children_def
thf(fact_6528_del__x__mi__lets__in__not__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_6529_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_6530_set__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A2 )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_6531_set__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_6532_set__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_6533_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_6534_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_6535_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_6536_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_6537_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X2: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
     => ( ( list_update_o @ Xs @ I @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_6538_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_6539_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_6540_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_6541_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_6542_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_6543_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_6544_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6545_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6546_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
          = ( set_o2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6547_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_6548_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
     => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_6549_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_6550_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_6551_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_6552_list__update__code_I2_J,axiom,
    ! [X2: int,Xs: list_int,Y3: int] :
      ( ( list_update_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
      = ( cons_int @ Y3 @ Xs ) ) ).

% list_update_code(2)
thf(fact_6553_list__update__code_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Y3: nat] :
      ( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
      = ( cons_nat @ Y3 @ Xs ) ) ).

% list_update_code(2)
thf(fact_6554_list__update__code_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
      = ( cons_VEBT_VEBT @ Y3 @ Xs ) ) ).

% list_update_code(2)
thf(fact_6555_set__update__subsetI,axiom,
    ! [Xs: list_real,A3: set_real,X2: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_6556_set__update__subsetI,axiom,
    ! [Xs: list_o,A3: set_o,X2: $o,I: nat] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
     => ( ( member_o @ X2 @ A3 )
       => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_6557_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A3: set_set_nat,X2: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A3 )
     => ( ( member_set_nat @ X2 @ A3 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_6558_set__update__subsetI,axiom,
    ! [Xs: list_nat,A3: set_nat,X2: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
     => ( ( member_nat @ X2 @ A3 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_6559_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
     => ( ( member_VEBT_VEBT @ X2 @ A3 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_6560_set__update__subsetI,axiom,
    ! [Xs: list_int,A3: set_int,X2: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
     => ( ( member_int @ X2 @ A3 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_6561_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X2 ) ) @ ( insert8211810215607154385at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_6562_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X2: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X2 ) ) @ ( insert_real @ X2 @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_6563_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,X2: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X2 ) ) @ ( insert_o @ X2 @ ( set_o2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_6564_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ ( insert_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_6565_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) ) @ ( insert_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_6566_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X2 ) ) @ ( insert_int @ X2 @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_6567_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X2: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_6568_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X2: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ X2 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_6569_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X2: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_6570_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_6571_set__update__memI,axiom,
    ! [N: nat,Xs: list_o,X2: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_6572_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_6573_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_6574_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ J )
            = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_6575_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X2: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X2 ) @ J )
        = ( ( ( I = J )
           => X2 )
          & ( ( I != J )
           => ( nth_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_6576_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_6577_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X2 )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_6578_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_6579_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ( list_update_o @ Xs @ I @ X2 )
          = Xs )
        = ( ( nth_o @ Xs @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_6580_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X2 )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_6581_insert__simp__norm,axiom,
    ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_6582_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X2: 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 @ X2 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( 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_6583_sup__bot__left,axiom,
    ! [X2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_6584_sup__bot__left,axiom,
    ! [X2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_6585_sup__bot__left,axiom,
    ! [X2: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_6586_sup__bot__left,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_6587_sup__bot__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_6588_sup__bot__left,axiom,
    ! [X2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X2 )
      = X2 ) ).

% sup_bot_left
thf(fact_6589_sup__bot__right,axiom,
    ! [X2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ X2 @ bot_bo5327735625951526323at_nat )
      = X2 ) ).

% sup_bot_right
thf(fact_6590_sup__bot__right,axiom,
    ! [X2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X2 @ bot_bo228742789529271731at_nat )
      = X2 ) ).

% sup_bot_right
thf(fact_6591_sup__bot__right,axiom,
    ! [X2: set_real] :
      ( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
      = X2 ) ).

% sup_bot_right
thf(fact_6592_sup__bot__right,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
      = X2 ) ).

% sup_bot_right
thf(fact_6593_sup__bot__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
      = X2 ) ).

% sup_bot_right
thf(fact_6594_sup__bot__right,axiom,
    ! [X2: set_int] :
      ( ( sup_sup_set_int @ X2 @ bot_bot_set_int )
      = X2 ) ).

% sup_bot_right
thf(fact_6595_bot__eq__sup__iff,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat
        = ( sup_su718114333110466843at_nat @ X2 @ Y3 ) )
      = ( ( X2 = bot_bo5327735625951526323at_nat )
        & ( Y3 = bot_bo5327735625951526323at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_6596_bot__eq__sup__iff,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) )
      = ( ( X2 = bot_bo228742789529271731at_nat )
        & ( Y3 = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_6597_bot__eq__sup__iff,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ X2 @ Y3 ) )
      = ( ( X2 = bot_bot_set_real )
        & ( Y3 = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
thf(fact_6598_bot__eq__sup__iff,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X2 @ Y3 ) )
      = ( ( X2 = bot_bot_set_o )
        & ( Y3 = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_6599_bot__eq__sup__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X2 @ Y3 ) )
      = ( ( X2 = bot_bot_set_nat )
        & ( Y3 = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_6600_bot__eq__sup__iff,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X2 @ Y3 ) )
      = ( ( X2 = bot_bot_set_int )
        & ( Y3 = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_6601_sup__eq__bot__iff,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
        = bot_bo5327735625951526323at_nat )
      = ( ( X2 = bot_bo5327735625951526323at_nat )
        & ( Y3 = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_6602_sup__eq__bot__iff,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
        = bot_bo228742789529271731at_nat )
      = ( ( X2 = bot_bo228742789529271731at_nat )
        & ( Y3 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_6603_sup__eq__bot__iff,axiom,
    ! [X2: set_real,Y3: set_real] :
      ( ( ( sup_sup_set_real @ X2 @ Y3 )
        = bot_bot_set_real )
      = ( ( X2 = bot_bot_set_real )
        & ( Y3 = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
thf(fact_6604_sup__eq__bot__iff,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( ( sup_sup_set_o @ X2 @ Y3 )
        = bot_bot_set_o )
      = ( ( X2 = bot_bot_set_o )
        & ( Y3 = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_6605_sup__eq__bot__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( sup_sup_set_nat @ X2 @ Y3 )
        = bot_bot_set_nat )
      = ( ( X2 = bot_bot_set_nat )
        & ( Y3 = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_6606_sup__eq__bot__iff,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ( sup_sup_set_int @ X2 @ Y3 )
        = bot_bot_set_int )
      = ( ( X2 = bot_bot_set_int )
        & ( Y3 = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_6607_le__inf__iff,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
      = ( ( ord_less_eq_set_nat @ X2 @ Y3 )
        & ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_6608_le__inf__iff,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) )
      = ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
        & ( ord_le3146513528884898305at_nat @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_6609_le__inf__iff,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) )
      = ( ( ord_less_eq_set_int @ X2 @ Y3 )
        & ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_6610_le__inf__iff,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) )
      = ( ( ord_less_eq_rat @ X2 @ Y3 )
        & ( ord_less_eq_rat @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_6611_le__inf__iff,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) )
      = ( ( ord_less_eq_nat @ X2 @ Y3 )
        & ( ord_less_eq_nat @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_6612_le__inf__iff,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) )
      = ( ( ord_less_eq_int @ X2 @ Y3 )
        & ( ord_less_eq_int @ X2 @ Z ) ) ) ).

% le_inf_iff
thf(fact_6613_inf_Obounded__iff,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
      = ( ( ord_less_eq_set_nat @ A2 @ B3 )
        & ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_6614_inf_Obounded__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
      = ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
        & ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_6615_inf_Obounded__iff,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) )
      = ( ( ord_less_eq_set_int @ A2 @ B3 )
        & ( ord_less_eq_set_int @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_6616_inf_Obounded__iff,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
      = ( ( ord_less_eq_rat @ A2 @ B3 )
        & ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_6617_inf_Obounded__iff,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
      = ( ( ord_less_eq_nat @ A2 @ B3 )
        & ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_6618_inf_Obounded__iff,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
      = ( ( ord_less_eq_int @ A2 @ B3 )
        & ( ord_less_eq_int @ A2 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_6619_le__sup__iff,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_eq_set_nat @ X2 @ Z )
        & ( ord_less_eq_set_nat @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6620_le__sup__iff,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) @ Z )
      = ( ( ord_le3000389064537975527at_nat @ X2 @ Z )
        & ( ord_le3000389064537975527at_nat @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6621_le__sup__iff,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) @ Z )
      = ( ( ord_le1268244103169919719at_nat @ X2 @ Z )
        & ( ord_le1268244103169919719at_nat @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6622_le__sup__iff,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_eq_set_int @ X2 @ Z )
        & ( ord_less_eq_set_int @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6623_le__sup__iff,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_eq_rat @ X2 @ Z )
        & ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6624_le__sup__iff,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_eq_nat @ X2 @ Z )
        & ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6625_le__sup__iff,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_eq_int @ X2 @ Z )
        & ( ord_less_eq_int @ Y3 @ Z ) ) ) ).

% le_sup_iff
thf(fact_6626_sup_Obounded__iff,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_set_nat @ B3 @ A2 )
        & ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6627_sup_Obounded__iff,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
      = ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
        & ( ord_le3000389064537975527at_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6628_sup_Obounded__iff,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
      = ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
        & ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6629_sup_Obounded__iff,axiom,
    ! [B3: set_int,C: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_set_int @ B3 @ A2 )
        & ( ord_less_eq_set_int @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6630_sup_Obounded__iff,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_rat @ B3 @ A2 )
        & ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6631_sup_Obounded__iff,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_nat @ B3 @ A2 )
        & ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6632_sup_Obounded__iff,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_int @ B3 @ A2 )
        & ( ord_less_eq_int @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_6633_inf__bot__left,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X2 )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_6634_inf__bot__left,axiom,
    ! [X2: set_real] :
      ( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
      = bot_bot_set_real ) ).

% inf_bot_left
thf(fact_6635_inf__bot__left,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
      = bot_bot_set_o ) ).

% inf_bot_left
thf(fact_6636_inf__bot__left,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_6637_inf__bot__left,axiom,
    ! [X2: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X2 )
      = bot_bot_set_int ) ).

% inf_bot_left
thf(fact_6638_inf__bot__right,axiom,
    ! [X2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X2 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_6639_inf__bot__right,axiom,
    ! [X2: set_real] :
      ( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% inf_bot_right
thf(fact_6640_inf__bot__right,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% inf_bot_right
thf(fact_6641_inf__bot__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_6642_inf__bot__right,axiom,
    ! [X2: set_int] :
      ( ( inf_inf_set_int @ X2 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% inf_bot_right
thf(fact_6643_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ A2 @ bot_bo5327735625951526323at_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_6644_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_6645_sup__bot_Oright__neutral,axiom,
    ! [A2: set_real] :
      ( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_6646_sup__bot_Oright__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_6647_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_6648_sup__bot_Oright__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_6649_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat
        = ( sup_su718114333110466843at_nat @ A2 @ B3 ) )
      = ( ( A2 = bot_bo5327735625951526323at_nat )
        & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_6650_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_6651_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ A2 @ B3 ) )
      = ( ( A2 = bot_bot_set_real )
        & ( B3 = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_6652_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B3: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A2 @ B3 ) )
      = ( ( A2 = bot_bot_set_o )
        & ( B3 = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_6653_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A2 @ B3 ) )
      = ( ( A2 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_6654_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A2 @ B3 ) )
      = ( ( A2 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_6655_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_6656_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_6657_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_6658_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_6659_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_6660_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_6661_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
        = bot_bo5327735625951526323at_nat )
      = ( ( A2 = bot_bo5327735625951526323at_nat )
        & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_6662_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
        = bot_bo228742789529271731at_nat )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_6663_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( ( sup_sup_set_real @ A2 @ B3 )
        = bot_bot_set_real )
      = ( ( A2 = bot_bot_set_real )
        & ( B3 = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_6664_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B3: set_o] :
      ( ( ( sup_sup_set_o @ A2 @ B3 )
        = bot_bot_set_o )
      = ( ( A2 = bot_bot_set_o )
        & ( B3 = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_6665_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B3 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_6666_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ( sup_sup_set_int @ A2 @ B3 )
        = bot_bot_set_int )
      = ( ( A2 = bot_bot_set_int )
        & ( B3 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_6667_max_Obounded__iff,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_rat @ B3 @ A2 )
        & ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_6668_max_Obounded__iff,axiom,
    ! [B3: num,C: num,A2: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_num @ B3 @ A2 )
        & ( ord_less_eq_num @ C @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_6669_max_Obounded__iff,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_nat @ B3 @ A2 )
        & ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_6670_max_Obounded__iff,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 )
      = ( ( ord_less_eq_int @ B3 @ A2 )
        & ( ord_less_eq_int @ C @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_6671_max_Oabsorb2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_max_rat @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_6672_max_Oabsorb2,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_eq_num @ A2 @ B3 )
     => ( ( ord_max_num @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_6673_max_Oabsorb2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_max_nat @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_6674_max_Oabsorb2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_max_int @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_6675_max_Oabsorb1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_max_rat @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb1
thf(fact_6676_max_Oabsorb1,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_eq_num @ B3 @ A2 )
     => ( ( ord_max_num @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb1
thf(fact_6677_max_Oabsorb1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_max_nat @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb1
thf(fact_6678_max_Oabsorb1,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_max_int @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb1
thf(fact_6679_max__less__iff__conj,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_real @ X2 @ Z )
        & ( ord_less_real @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_6680_max__less__iff__conj,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_rat @ X2 @ Z )
        & ( ord_less_rat @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_6681_max__less__iff__conj,axiom,
    ! [X2: num,Y3: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_num @ X2 @ Z )
        & ( ord_less_num @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_6682_max__less__iff__conj,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_nat @ X2 @ Z )
        & ( ord_less_nat @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_6683_max__less__iff__conj,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
      = ( ( ord_less_int @ X2 @ Z )
        & ( ord_less_int @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_6684_max_Oabsorb4,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ord_max_real @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_6685_max_Oabsorb4,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( ord_max_rat @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_6686_max_Oabsorb4,axiom,
    ! [A2: num,B3: num] :
      ( ( ord_less_num @ A2 @ B3 )
     => ( ( ord_max_num @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_6687_max_Oabsorb4,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_max_nat @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_6688_max_Oabsorb4,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_max_int @ A2 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_6689_max_Oabsorb3,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( ord_max_real @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb3
thf(fact_6690_max_Oabsorb3,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( ord_max_rat @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb3
thf(fact_6691_max_Oabsorb3,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_num @ B3 @ A2 )
     => ( ( ord_max_num @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb3
thf(fact_6692_max_Oabsorb3,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( ord_max_nat @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb3
thf(fact_6693_max_Oabsorb3,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( ord_max_int @ A2 @ B3 )
        = A2 ) ) ).

% max.absorb3
thf(fact_6694_max__bot,axiom,
    ! [X2: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X2 )
      = X2 ) ).

% max_bot
thf(fact_6695_max__bot,axiom,
    ! [X2: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X2 )
      = X2 ) ).

% max_bot
thf(fact_6696_max__bot,axiom,
    ! [X2: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X2 )
      = X2 ) ).

% max_bot
thf(fact_6697_max__bot,axiom,
    ! [X2: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X2 )
      = X2 ) ).

% max_bot
thf(fact_6698_max__bot,axiom,
    ! [X2: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X2 )
      = X2 ) ).

% max_bot
thf(fact_6699_max__bot2,axiom,
    ! [X2: set_real] :
      ( ( ord_max_set_real @ X2 @ bot_bot_set_real )
      = X2 ) ).

% max_bot2
thf(fact_6700_max__bot2,axiom,
    ! [X2: set_o] :
      ( ( ord_max_set_o @ X2 @ bot_bot_set_o )
      = X2 ) ).

% max_bot2
thf(fact_6701_max__bot2,axiom,
    ! [X2: set_nat] :
      ( ( ord_max_set_nat @ X2 @ bot_bot_set_nat )
      = X2 ) ).

% max_bot2
thf(fact_6702_max__bot2,axiom,
    ! [X2: set_int] :
      ( ( ord_max_set_int @ X2 @ bot_bot_set_int )
      = X2 ) ).

% max_bot2
thf(fact_6703_max__bot2,axiom,
    ! [X2: nat] :
      ( ( ord_max_nat @ X2 @ bot_bot_nat )
      = X2 ) ).

% max_bot2
thf(fact_6704_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_6705_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( ord_max_nat @ A2 @ B3 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_6706_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% max_nat.left_neutral
thf(fact_6707_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A2 @ B3 ) )
      = ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_6708_max__nat_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% max_nat.right_neutral
thf(fact_6709_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_6710_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_6711_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_6712_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_6713_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ V ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_6714_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_6715_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_6716_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_6717_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_6718_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(4)
thf(fact_6719_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(4)
thf(fact_6720_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(4)
thf(fact_6721_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(4)
thf(fact_6722_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(4)
thf(fact_6723_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(4)
thf(fact_6724_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(3)
thf(fact_6725_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(3)
thf(fact_6726_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(3)
thf(fact_6727_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(3)
thf(fact_6728_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(3)
thf(fact_6729_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X2 ) )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(3)
thf(fact_6730_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_6731_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_6732_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_6733_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_6734_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_6735_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_6736_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_6737_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_6738_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6739_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6740_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6741_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6742_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6743_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6744_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6745_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6746_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6747_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6748_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6749_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6750_max_OcoboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C @ B3 )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_6751_max_OcoboundedI2,axiom,
    ! [C: num,B3: num,A2: num] :
      ( ( ord_less_eq_num @ C @ B3 )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_6752_max_OcoboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ B3 )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_6753_max_OcoboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( ord_less_eq_int @ C @ B3 )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_6754_max_OcoboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ A2 )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_6755_max_OcoboundedI1,axiom,
    ! [C: num,A2: num,B3: num] :
      ( ( ord_less_eq_num @ C @ A2 )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_6756_max_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_6757_max_OcoboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_6758_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_6759_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_6760_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_6761_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_6762_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_max_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_6763_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_max_num @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_6764_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_max_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_6765_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_max_int @ A4 @ B4 )
          = A4 ) ) ) ).

% max.absorb_iff1
thf(fact_6766_le__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
      = ( ( ord_less_eq_rat @ Z @ X2 )
        | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_6767_le__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y3: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
      = ( ( ord_less_eq_num @ Z @ X2 )
        | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_6768_le__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
      = ( ( ord_less_eq_nat @ Z @ X2 )
        | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_6769_le__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
      = ( ( ord_less_eq_int @ Z @ X2 )
        | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_6770_max_Ocobounded2,axiom,
    ! [B3: rat,A2: rat] : ( ord_less_eq_rat @ B3 @ ( ord_max_rat @ A2 @ B3 ) ) ).

% max.cobounded2
thf(fact_6771_max_Ocobounded2,axiom,
    ! [B3: num,A2: num] : ( ord_less_eq_num @ B3 @ ( ord_max_num @ A2 @ B3 ) ) ).

% max.cobounded2
thf(fact_6772_max_Ocobounded2,axiom,
    ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( ord_max_nat @ A2 @ B3 ) ) ).

% max.cobounded2
thf(fact_6773_max_Ocobounded2,axiom,
    ! [B3: int,A2: int] : ( ord_less_eq_int @ B3 @ ( ord_max_int @ A2 @ B3 ) ) ).

% max.cobounded2
thf(fact_6774_max_Ocobounded1,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ A2 @ ( ord_max_rat @ A2 @ B3 ) ) ).

% max.cobounded1
thf(fact_6775_max_Ocobounded1,axiom,
    ! [A2: num,B3: num] : ( ord_less_eq_num @ A2 @ ( ord_max_num @ A2 @ B3 ) ) ).

% max.cobounded1
thf(fact_6776_max_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( ord_max_nat @ A2 @ B3 ) ) ).

% max.cobounded1
thf(fact_6777_max_Ocobounded1,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ A2 @ ( ord_max_int @ A2 @ B3 ) ) ).

% max.cobounded1
thf(fact_6778_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( A4
          = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_6779_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( A4
          = ( ord_max_num @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_6780_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_6781_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( A4
          = ( ord_max_int @ A4 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_6782_max_OboundedI,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ C @ A2 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_6783_max_OboundedI,axiom,
    ! [B3: num,A2: num,C: num] :
      ( ( ord_less_eq_num @ B3 @ A2 )
     => ( ( ord_less_eq_num @ C @ A2 )
       => ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_6784_max_OboundedI,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_6785_max_OboundedI,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ C @ A2 )
       => ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_6786_max_OboundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_rat @ B3 @ A2 )
         => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% max.boundedE
thf(fact_6787_max_OboundedE,axiom,
    ! [B3: num,C: num,A2: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_num @ B3 @ A2 )
         => ~ ( ord_less_eq_num @ C @ A2 ) ) ) ).

% max.boundedE
thf(fact_6788_max_OboundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
         => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% max.boundedE
thf(fact_6789_max_OboundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_int @ B3 @ A2 )
         => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).

% max.boundedE
thf(fact_6790_max_OorderI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( ord_max_rat @ A2 @ B3 ) )
     => ( ord_less_eq_rat @ B3 @ A2 ) ) ).

% max.orderI
thf(fact_6791_max_OorderI,axiom,
    ! [A2: num,B3: num] :
      ( ( A2
        = ( ord_max_num @ A2 @ B3 ) )
     => ( ord_less_eq_num @ B3 @ A2 ) ) ).

% max.orderI
thf(fact_6792_max_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2
        = ( ord_max_nat @ A2 @ B3 ) )
     => ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% max.orderI
thf(fact_6793_max_OorderI,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( ord_max_int @ A2 @ B3 ) )
     => ( ord_less_eq_int @ B3 @ A2 ) ) ).

% max.orderI
thf(fact_6794_max_OorderE,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( A2
        = ( ord_max_rat @ A2 @ B3 ) ) ) ).

% max.orderE
thf(fact_6795_max_OorderE,axiom,
    ! [B3: num,A2: num] :
      ( ( ord_less_eq_num @ B3 @ A2 )
     => ( A2
        = ( ord_max_num @ A2 @ B3 ) ) ) ).

% max.orderE
thf(fact_6796_max_OorderE,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( A2
        = ( ord_max_nat @ A2 @ B3 ) ) ) ).

% max.orderE
thf(fact_6797_max_OorderE,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( A2
        = ( ord_max_int @ A2 @ B3 ) ) ) ).

% max.orderE
thf(fact_6798_max_Omono,axiom,
    ! [C: rat,A2: rat,D: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ A2 )
     => ( ( ord_less_eq_rat @ D @ B3 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A2 @ B3 ) ) ) ) ).

% max.mono
thf(fact_6799_max_Omono,axiom,
    ! [C: num,A2: num,D: num,B3: num] :
      ( ( ord_less_eq_num @ C @ A2 )
     => ( ( ord_less_eq_num @ D @ B3 )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A2 @ B3 ) ) ) ) ).

% max.mono
thf(fact_6800_max_Omono,axiom,
    ! [C: nat,A2: nat,D: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ( ord_less_eq_nat @ D @ B3 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A2 @ B3 ) ) ) ) ).

% max.mono
thf(fact_6801_max_Omono,axiom,
    ! [C: int,A2: int,D: int,B3: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ( ord_less_eq_int @ D @ B3 )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A2 @ B3 ) ) ) ) ).

% max.mono
thf(fact_6802_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_6803_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M3 @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_6804_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_6805_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_6806_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y3 ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).

% of_nat_max
thf(fact_6807_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y3 ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).

% of_nat_max
thf(fact_6808_of__nat__max,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y3 ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).

% of_nat_max
thf(fact_6809_less__max__iff__disj,axiom,
    ! [Z: real,X2: real,Y3: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y3 ) )
      = ( ( ord_less_real @ Z @ X2 )
        | ( ord_less_real @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_6810_less__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y3: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
      = ( ( ord_less_rat @ Z @ X2 )
        | ( ord_less_rat @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_6811_less__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y3: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
      = ( ( ord_less_num @ Z @ X2 )
        | ( ord_less_num @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_6812_less__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y3: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
      = ( ( ord_less_nat @ Z @ X2 )
        | ( ord_less_nat @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_6813_less__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y3: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
      = ( ( ord_less_int @ Z @ X2 )
        | ( ord_less_int @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_6814_max_Ostrict__boundedE,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_real @ ( ord_max_real @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_real @ B3 @ A2 )
         => ~ ( ord_less_real @ C @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_6815_max_Ostrict__boundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_rat @ B3 @ A2 )
         => ~ ( ord_less_rat @ C @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_6816_max_Ostrict__boundedE,axiom,
    ! [B3: num,C: num,A2: num] :
      ( ( ord_less_num @ ( ord_max_num @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_num @ B3 @ A2 )
         => ~ ( ord_less_num @ C @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_6817_max_Ostrict__boundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_nat @ B3 @ A2 )
         => ~ ( ord_less_nat @ C @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_6818_max_Ostrict__boundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_int @ ( ord_max_int @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_int @ B3 @ A2 )
         => ~ ( ord_less_int @ C @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_6819_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( A4
            = ( ord_max_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_6820_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( A4
            = ( ord_max_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_6821_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( A4
            = ( ord_max_num @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_6822_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( ord_max_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_6823_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( A4
            = ( ord_max_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_6824_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ A2 )
     => ( ord_less_real @ C @ ( ord_max_real @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_6825_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ A2 )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_6826_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A2: num,B3: num] :
      ( ( ord_less_num @ C @ A2 )
     => ( ord_less_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_6827_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ C @ A2 )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_6828_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ C @ A2 )
     => ( ord_less_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_6829_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ C @ B3 )
     => ( ord_less_real @ C @ ( ord_max_real @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_6830_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ C @ B3 )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_6831_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B3: num,A2: num] :
      ( ( ord_less_num @ C @ B3 )
     => ( ord_less_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_6832_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ord_less_nat @ C @ B3 )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_6833_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( ord_less_int @ C @ B3 )
     => ( ord_less_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_6834_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_6835_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_6836_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_6837_max__def,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_6838_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_6839_max__def,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def
thf(fact_6840_max__absorb1,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X2 )
     => ( ( ord_max_set_int @ X2 @ Y3 )
        = X2 ) ) ).

% max_absorb1
thf(fact_6841_max__absorb1,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ( ord_max_rat @ X2 @ Y3 )
        = X2 ) ) ).

% max_absorb1
thf(fact_6842_max__absorb1,axiom,
    ! [Y3: num,X2: num] :
      ( ( ord_less_eq_num @ Y3 @ X2 )
     => ( ( ord_max_num @ X2 @ Y3 )
        = X2 ) ) ).

% max_absorb1
thf(fact_6843_max__absorb1,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( ord_max_nat @ X2 @ Y3 )
        = X2 ) ) ).

% max_absorb1
thf(fact_6844_max__absorb1,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ Y3 @ X2 )
     => ( ( ord_max_int @ X2 @ Y3 )
        = X2 ) ) ).

% max_absorb1
thf(fact_6845_max__absorb2,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ord_max_set_int @ X2 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_6846_max__absorb2,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ord_max_rat @ X2 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_6847_max__absorb2,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_num @ X2 @ Y3 )
     => ( ( ord_max_num @ X2 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_6848_max__absorb2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_max_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_6849_max__absorb2,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ord_max_int @ X2 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_6850_max__add__distrib__right,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y3 @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_6851_max__add__distrib__right,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y3 @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_6852_max__add__distrib__right,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y3 @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_6853_max__add__distrib__right,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y3 @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_6854_max__add__distrib__left,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_6855_max__add__distrib__left,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_6856_max__add__distrib__left,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_6857_max__add__distrib__left,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_6858_max__diff__distrib__left,axiom,
    ! [X2: real,Y3: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_6859_max__diff__distrib__left,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X2 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_6860_max__diff__distrib__left,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_6861_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_6862_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_6863_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_6864_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_6865_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_6866_inf__sup__ord_I2_J,axiom,
    ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_6867_inf__sup__ord_I2_J,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_6868_inf__sup__ord_I2_J,axiom,
    ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_6869_inf__sup__ord_I2_J,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_6870_inf__sup__ord_I2_J,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_6871_inf__sup__ord_I2_J,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_6872_inf__sup__ord_I1_J,axiom,
    ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_6873_inf__sup__ord_I1_J,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_6874_inf__sup__ord_I1_J,axiom,
    ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_6875_inf__sup__ord_I1_J,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_6876_inf__sup__ord_I1_J,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_6877_inf__sup__ord_I1_J,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_6878_inf__le1,axiom,
    ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).

% inf_le1
thf(fact_6879_inf__le1,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ X2 ) ).

% inf_le1
thf(fact_6880_inf__le1,axiom,
    ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ X2 ) ).

% inf_le1
thf(fact_6881_inf__le1,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ X2 ) ).

% inf_le1
thf(fact_6882_inf__le1,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).

% inf_le1
thf(fact_6883_inf__le1,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ X2 ) ).

% inf_le1
thf(fact_6884_inf__le2,axiom,
    ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_6885_inf__le2,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_6886_inf__le2,axiom,
    ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_6887_inf__le2,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_6888_inf__le2,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_6889_inf__le2,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_6890_le__infE,axiom,
    ! [X2: set_nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) )
     => ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
         => ~ ( ord_less_eq_set_nat @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_6891_le__infE,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X2 @ A2 )
         => ~ ( ord_le3146513528884898305at_nat @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_6892_le__infE,axiom,
    ! [X2: set_int,A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ A2 @ B3 ) )
     => ~ ( ( ord_less_eq_set_int @ X2 @ A2 )
         => ~ ( ord_less_eq_set_int @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_6893_le__infE,axiom,
    ! [X2: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B3 ) )
     => ~ ( ( ord_less_eq_rat @ X2 @ A2 )
         => ~ ( ord_less_eq_rat @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_6894_le__infE,axiom,
    ! [X2: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) )
     => ~ ( ( ord_less_eq_nat @ X2 @ A2 )
         => ~ ( ord_less_eq_nat @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_6895_le__infE,axiom,
    ! [X2: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A2 @ B3 ) )
     => ~ ( ( ord_less_eq_int @ X2 @ A2 )
         => ~ ( ord_less_eq_int @ X2 @ B3 ) ) ) ).

% le_infE
thf(fact_6896_le__infI,axiom,
    ! [X2: set_nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ A2 )
     => ( ( ord_less_eq_set_nat @ X2 @ B3 )
       => ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).

% le_infI
thf(fact_6897_le__infI,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X2 @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ X2 @ B3 )
       => ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) ) ) ) ).

% le_infI
thf(fact_6898_le__infI,axiom,
    ! [X2: set_int,A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ A2 )
     => ( ( ord_less_eq_set_int @ X2 @ B3 )
       => ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ A2 @ B3 ) ) ) ) ).

% le_infI
thf(fact_6899_le__infI,axiom,
    ! [X2: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ X2 @ A2 )
     => ( ( ord_less_eq_rat @ X2 @ B3 )
       => ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B3 ) ) ) ) ).

% le_infI
thf(fact_6900_le__infI,axiom,
    ! [X2: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X2 @ A2 )
     => ( ( ord_less_eq_nat @ X2 @ B3 )
       => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) ) ) ) ).

% le_infI
thf(fact_6901_le__infI,axiom,
    ! [X2: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ X2 @ A2 )
     => ( ( ord_less_eq_int @ X2 @ B3 )
       => ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A2 @ B3 ) ) ) ) ).

% le_infI
thf(fact_6902_inf__mono,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ( ord_less_eq_set_nat @ B3 @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_6903_inf__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ C )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ D )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ ( inf_in2572325071724192079at_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_6904_inf__mono,axiom,
    ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C )
     => ( ( ord_less_eq_set_int @ B3 @ D )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ ( inf_inf_set_int @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_6905_inf__mono,axiom,
    ! [A2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ C )
     => ( ( ord_less_eq_rat @ B3 @ D )
       => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ ( inf_inf_rat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_6906_inf__mono,axiom,
    ! [A2: nat,C: nat,B3: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B3 @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_6907_inf__mono,axiom,
    ! [A2: int,C: int,B3: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ( ord_less_eq_int @ B3 @ D )
       => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ ( inf_inf_int @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_6908_le__infI1,axiom,
    ! [A2: set_nat,X2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_6909_le__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ X2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_6910_le__infI1,axiom,
    ! [A2: set_int,X2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ X2 )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_6911_le__infI1,axiom,
    ! [A2: rat,X2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ X2 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_6912_le__infI1,axiom,
    ! [A2: nat,X2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ X2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_6913_le__infI1,axiom,
    ! [A2: int,X2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ X2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).

% le_infI1
thf(fact_6914_le__infI2,axiom,
    ! [B3: set_nat,X2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ X2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_6915_le__infI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ X2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_6916_le__infI2,axiom,
    ! [B3: set_int,X2: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ X2 )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_6917_le__infI2,axiom,
    ! [B3: rat,X2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ X2 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_6918_le__infI2,axiom,
    ! [B3: nat,X2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ X2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_6919_le__infI2,axiom,
    ! [B3: int,X2: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ X2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).

% le_infI2
thf(fact_6920_inf_OorderE,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( A2
        = ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).

% inf.orderE
thf(fact_6921_inf_OorderE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
     => ( A2
        = ( inf_in2572325071724192079at_nat @ A2 @ B3 ) ) ) ).

% inf.orderE
thf(fact_6922_inf_OorderE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( A2
        = ( inf_inf_set_int @ A2 @ B3 ) ) ) ).

% inf.orderE
thf(fact_6923_inf_OorderE,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( A2
        = ( inf_inf_rat @ A2 @ B3 ) ) ) ).

% inf.orderE
thf(fact_6924_inf_OorderE,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( A2
        = ( inf_inf_nat @ A2 @ B3 ) ) ) ).

% inf.orderE
thf(fact_6925_inf_OorderE,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( A2
        = ( inf_inf_int @ A2 @ B3 ) ) ) ).

% inf.orderE
thf(fact_6926_inf_OorderI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( A2
        = ( inf_inf_set_nat @ A2 @ B3 ) )
     => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).

% inf.orderI
thf(fact_6927_inf_OorderI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( A2
        = ( inf_in2572325071724192079at_nat @ A2 @ B3 ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B3 ) ) ).

% inf.orderI
thf(fact_6928_inf_OorderI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2
        = ( inf_inf_set_int @ A2 @ B3 ) )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% inf.orderI
thf(fact_6929_inf_OorderI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( inf_inf_rat @ A2 @ B3 ) )
     => ( ord_less_eq_rat @ A2 @ B3 ) ) ).

% inf.orderI
thf(fact_6930_inf_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2
        = ( inf_inf_nat @ A2 @ B3 ) )
     => ( ord_less_eq_nat @ A2 @ B3 ) ) ).

% inf.orderI
thf(fact_6931_inf_OorderI,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( inf_inf_int @ A2 @ B3 ) )
     => ( ord_less_eq_int @ A2 @ B3 ) ) ).

% inf.orderI
thf(fact_6932_inf__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
      ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
              ( ( ord_less_eq_set_nat @ X5 @ Y4 )
             => ( ( ord_less_eq_set_nat @ X5 @ Z4 )
               => ( ord_less_eq_set_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_set_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_6933_inf__unique,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X5 @ Y4 )
             => ( ( ord_le3146513528884898305at_nat @ X5 @ Z4 )
               => ( ord_le3146513528884898305at_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_6934_inf__unique,axiom,
    ! [F: set_int > set_int > set_int,X2: set_int,Y3: set_int] :
      ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: set_int,Y4: set_int,Z4: set_int] :
              ( ( ord_less_eq_set_int @ X5 @ Y4 )
             => ( ( ord_less_eq_set_int @ X5 @ Z4 )
               => ( ord_less_eq_set_int @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_set_int @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_6935_inf__unique,axiom,
    ! [F: rat > rat > rat,X2: rat,Y3: rat] :
      ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: rat,Y4: rat,Z4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ( ord_less_eq_rat @ X5 @ Z4 )
               => ( ord_less_eq_rat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_rat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_6936_inf__unique,axiom,
    ! [F: nat > nat > nat,X2: nat,Y3: nat] :
      ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: nat,Y4: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ( ord_less_eq_nat @ X5 @ Z4 )
               => ( ord_less_eq_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_6937_inf__unique,axiom,
    ! [F: int > int > int,X2: int,Y3: int] :
      ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: int,Y4: int,Z4: int] :
              ( ( ord_less_eq_int @ X5 @ Y4 )
             => ( ( ord_less_eq_int @ X5 @ Z4 )
               => ( ord_less_eq_int @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
         => ( ( inf_inf_int @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_6938_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( inf_inf_set_nat @ X @ Y )
          = X ) ) ) ).

% le_iff_inf
thf(fact_6939_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X @ Y )
          = X ) ) ) ).

% le_iff_inf
thf(fact_6940_le__iff__inf,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( inf_inf_set_int @ X @ Y )
          = X ) ) ) ).

% le_iff_inf
thf(fact_6941_le__iff__inf,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( inf_inf_rat @ X @ Y )
          = X ) ) ) ).

% le_iff_inf
thf(fact_6942_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( inf_inf_nat @ X @ Y )
          = X ) ) ) ).

% le_iff_inf
thf(fact_6943_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( inf_inf_int @ X @ Y )
          = X ) ) ) ).

% le_iff_inf
thf(fact_6944_inf_Oabsorb1,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( inf_inf_set_nat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_6945_inf_Oabsorb1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_6946_inf_Oabsorb1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( inf_inf_set_int @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_6947_inf_Oabsorb1,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( inf_inf_rat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_6948_inf_Oabsorb1,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( inf_inf_nat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_6949_inf_Oabsorb1,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( inf_inf_int @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb1
thf(fact_6950_inf_Oabsorb2,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_6951_inf_Oabsorb2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A2 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_6952_inf_Oabsorb2,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( inf_inf_set_int @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_6953_inf_Oabsorb2,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( inf_inf_rat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_6954_inf_Oabsorb2,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_6955_inf_Oabsorb2,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( inf_inf_int @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb2
thf(fact_6956_inf__absorb1,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y3 )
     => ( ( inf_inf_set_nat @ X2 @ Y3 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_6957_inf__absorb1,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
     => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_6958_inf__absorb1,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( inf_inf_set_int @ X2 @ Y3 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_6959_inf__absorb1,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( inf_inf_rat @ X2 @ Y3 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_6960_inf__absorb1,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( inf_inf_nat @ X2 @ Y3 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_6961_inf__absorb1,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( inf_inf_int @ X2 @ Y3 )
        = X2 ) ) ).

% inf_absorb1
thf(fact_6962_inf__absorb2,axiom,
    ! [Y3: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X2 )
     => ( ( inf_inf_set_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_6963_inf__absorb2,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ Y3 @ X2 )
     => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_6964_inf__absorb2,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X2 )
     => ( ( inf_inf_set_int @ X2 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_6965_inf__absorb2,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ( inf_inf_rat @ X2 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_6966_inf__absorb2,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( inf_inf_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_6967_inf__absorb2,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ Y3 @ X2 )
     => ( ( inf_inf_int @ X2 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_6968_inf_OboundedE,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
         => ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_6969_inf_OboundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
         => ~ ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_6970_inf_OboundedE,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ~ ( ord_less_eq_set_int @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_6971_inf_OboundedE,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
     => ~ ( ( ord_less_eq_rat @ A2 @ B3 )
         => ~ ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_6972_inf_OboundedE,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
     => ~ ( ( ord_less_eq_nat @ A2 @ B3 )
         => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_6973_inf_OboundedE,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
     => ~ ( ( ord_less_eq_int @ A2 @ B3 )
         => ~ ( ord_less_eq_int @ A2 @ C ) ) ) ).

% inf.boundedE
thf(fact_6974_inf_OboundedI,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ A2 @ C )
       => ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_6975_inf_OboundedI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ C )
       => ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_6976_inf_OboundedI,axiom,
    ! [A2: set_int,B3: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ C )
       => ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_6977_inf_OboundedI,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( ord_less_eq_rat @ A2 @ C )
       => ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_6978_inf_OboundedI,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ A2 @ C )
       => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_6979_inf_OboundedI,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_eq_int @ A2 @ C )
       => ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) ) ) ) ).

% inf.boundedI
thf(fact_6980_inf__greatest,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_set_nat @ X2 @ Z )
       => ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_6981_inf__greatest,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
     => ( ( ord_le3146513528884898305at_nat @ X2 @ Z )
       => ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_6982_inf__greatest,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( ord_less_eq_set_int @ X2 @ Z )
       => ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_6983_inf__greatest,axiom,
    ! [X2: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( ord_less_eq_rat @ X2 @ Z )
       => ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_6984_inf__greatest,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ X2 @ Z )
       => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_6985_inf__greatest,axiom,
    ! [X2: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( ord_less_eq_int @ X2 @ Z )
       => ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_6986_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( A4
          = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_6987_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( A4
          = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_6988_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( A4
          = ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_6989_inf_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( A4
          = ( inf_inf_rat @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_6990_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( A4
          = ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_6991_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( A4
          = ( inf_inf_int @ A4 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_6992_inf_Ocobounded1,axiom,
    ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ A2 ) ).

% inf.cobounded1
thf(fact_6993_inf_Ocobounded1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ A2 ) ).

% inf.cobounded1
thf(fact_6994_inf_Ocobounded1,axiom,
    ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ A2 ) ).

% inf.cobounded1
thf(fact_6995_inf_Ocobounded1,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ A2 ) ).

% inf.cobounded1
thf(fact_6996_inf_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ A2 ) ).

% inf.cobounded1
thf(fact_6997_inf_Ocobounded1,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ A2 ) ).

% inf.cobounded1
thf(fact_6998_inf_Ocobounded2,axiom,
    ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_6999_inf_Ocobounded2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_7000_inf_Ocobounded2,axiom,
    ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_7001_inf_Ocobounded2,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_7002_inf_Ocobounded2,axiom,
    ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_7003_inf_Ocobounded2,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ B3 ) ).

% inf.cobounded2
thf(fact_7004_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( inf_inf_set_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_7005_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_7006_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( inf_inf_set_int @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_7007_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( inf_inf_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_7008_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( inf_inf_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_7009_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( inf_inf_int @ A4 @ B4 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_7010_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( inf_inf_set_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_7011_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_7012_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( inf_inf_set_int @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_7013_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( inf_inf_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_7014_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( inf_inf_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_7015_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( inf_inf_int @ A4 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_7016_inf_OcoboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_7017_inf_OcoboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_7018_inf_OcoboundedI1,axiom,
    ! [A2: set_int,C: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_7019_inf_OcoboundedI1,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_7020_inf_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_7021_inf_OcoboundedI1,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI1
thf(fact_7022_inf_OcoboundedI2,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_7023_inf_OcoboundedI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_7024_inf_OcoboundedI2,axiom,
    ! [B3: set_int,C: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ C )
     => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_7025_inf_OcoboundedI2,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_7026_inf_OcoboundedI2,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_7027_inf_OcoboundedI2,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).

% inf.coboundedI2
thf(fact_7028_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7029_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7030_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7031_inf__sup__ord_I4_J,axiom,
    ! [Y3: set_int,X2: set_int] : ( ord_less_eq_set_int @ Y3 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7032_inf__sup__ord_I4_J,axiom,
    ! [Y3: rat,X2: rat] : ( ord_less_eq_rat @ Y3 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7033_inf__sup__ord_I4_J,axiom,
    ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7034_inf__sup__ord_I4_J,axiom,
    ! [Y3: int,X2: int] : ( ord_less_eq_int @ Y3 @ ( sup_sup_int @ X2 @ Y3 ) ) ).

% inf_sup_ord(4)
thf(fact_7035_inf__sup__ord_I3_J,axiom,
    ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7036_inf__sup__ord_I3_J,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7037_inf__sup__ord_I3_J,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7038_inf__sup__ord_I3_J,axiom,
    ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7039_inf__sup__ord_I3_J,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7040_inf__sup__ord_I3_J,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7041_inf__sup__ord_I3_J,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y3 ) ) ).

% inf_sup_ord(3)
thf(fact_7042_le__supE,axiom,
    ! [A2: set_nat,B3: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
         => ~ ( ord_less_eq_set_nat @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7043_le__supE,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_le3000389064537975527at_nat @ A2 @ X2 )
         => ~ ( ord_le3000389064537975527at_nat @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7044_le__supE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_le1268244103169919719at_nat @ A2 @ X2 )
         => ~ ( ord_le1268244103169919719at_nat @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7045_le__supE,axiom,
    ! [A2: set_int,B3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ X2 )
         => ~ ( ord_less_eq_set_int @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7046_le__supE,axiom,
    ! [A2: rat,B3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_less_eq_rat @ A2 @ X2 )
         => ~ ( ord_less_eq_rat @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7047_le__supE,axiom,
    ! [A2: nat,B3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_less_eq_nat @ A2 @ X2 )
         => ~ ( ord_less_eq_nat @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7048_le__supE,axiom,
    ! [A2: int,B3: int,X2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ X2 )
     => ~ ( ( ord_less_eq_int @ A2 @ X2 )
         => ~ ( ord_less_eq_int @ B3 @ X2 ) ) ) ).

% le_supE
thf(fact_7049_le__supI,axiom,
    ! [A2: set_nat,X2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X2 )
     => ( ( ord_less_eq_set_nat @ B3 @ X2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7050_le__supI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ X2 )
     => ( ( ord_le3000389064537975527at_nat @ B3 @ X2 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7051_le__supI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ X2 )
     => ( ( ord_le1268244103169919719at_nat @ B3 @ X2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7052_le__supI,axiom,
    ! [A2: set_int,X2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ X2 )
     => ( ( ord_less_eq_set_int @ B3 @ X2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7053_le__supI,axiom,
    ! [A2: rat,X2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ X2 )
     => ( ( ord_less_eq_rat @ B3 @ X2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7054_le__supI,axiom,
    ! [A2: nat,X2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ X2 )
     => ( ( ord_less_eq_nat @ B3 @ X2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7055_le__supI,axiom,
    ! [A2: int,X2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ X2 )
     => ( ( ord_less_eq_int @ B3 @ X2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ X2 ) ) ) ).

% le_supI
thf(fact_7056_sup__ge1,axiom,
    ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7057_sup__ge1,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7058_sup__ge1,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7059_sup__ge1,axiom,
    ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7060_sup__ge1,axiom,
    ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7061_sup__ge1,axiom,
    ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7062_sup__ge1,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y3 ) ) ).

% sup_ge1
thf(fact_7063_sup__ge2,axiom,
    ! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7064_sup__ge2,axiom,
    ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7065_sup__ge2,axiom,
    ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7066_sup__ge2,axiom,
    ! [Y3: set_int,X2: set_int] : ( ord_less_eq_set_int @ Y3 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7067_sup__ge2,axiom,
    ! [Y3: rat,X2: rat] : ( ord_less_eq_rat @ Y3 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7068_sup__ge2,axiom,
    ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7069_sup__ge2,axiom,
    ! [Y3: int,X2: int] : ( ord_less_eq_int @ Y3 @ ( sup_sup_int @ X2 @ Y3 ) ) ).

% sup_ge2
thf(fact_7070_le__supI1,axiom,
    ! [X2: set_nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ A2 )
     => ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7071_le__supI1,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ X2 @ A2 )
     => ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7072_le__supI1,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X2 @ A2 )
     => ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7073_le__supI1,axiom,
    ! [X2: set_int,A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ A2 )
     => ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7074_le__supI1,axiom,
    ! [X2: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ X2 @ A2 )
     => ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7075_le__supI1,axiom,
    ! [X2: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ X2 @ A2 )
     => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7076_le__supI1,axiom,
    ! [X2: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ X2 @ A2 )
     => ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% le_supI1
thf(fact_7077_le__supI2,axiom,
    ! [X2: set_nat,B3: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ B3 )
     => ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7078_le__supI2,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ X2 @ B3 )
     => ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7079_le__supI2,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X2 @ B3 )
     => ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7080_le__supI2,axiom,
    ! [X2: set_int,B3: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ B3 )
     => ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7081_le__supI2,axiom,
    ! [X2: rat,B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ X2 @ B3 )
     => ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7082_le__supI2,axiom,
    ! [X2: nat,B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X2 @ B3 )
     => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7083_le__supI2,axiom,
    ! [X2: int,B3: int,A2: int] :
      ( ( ord_less_eq_int @ X2 @ B3 )
     => ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% le_supI2
thf(fact_7084_sup_Omono,axiom,
    ! [C: set_nat,A2: set_nat,D: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A2 )
     => ( ( ord_less_eq_set_nat @ D @ B3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7085_sup_Omono,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C @ A2 )
     => ( ( ord_le3000389064537975527at_nat @ D @ B3 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ C @ D ) @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7086_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ D @ B3 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D ) @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7087_sup_Omono,axiom,
    ! [C: set_int,A2: set_int,D: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ C @ A2 )
     => ( ( ord_less_eq_set_int @ D @ B3 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C @ D ) @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7088_sup_Omono,axiom,
    ! [C: rat,A2: rat,D: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ A2 )
     => ( ( ord_less_eq_rat @ D @ B3 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7089_sup_Omono,axiom,
    ! [C: nat,A2: nat,D: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ( ord_less_eq_nat @ D @ B3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7090_sup_Omono,axiom,
    ! [C: int,A2: int,D: int,B3: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ( ord_less_eq_int @ D @ B3 )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A2 @ B3 ) ) ) ) ).

% sup.mono
thf(fact_7091_sup__mono,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ( ord_less_eq_set_nat @ B3 @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7092_sup__mono,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ C )
     => ( ( ord_le3000389064537975527at_nat @ B3 @ D )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ ( sup_su718114333110466843at_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7093_sup__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ C )
     => ( ( ord_le1268244103169919719at_nat @ B3 @ D )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ ( sup_su5525570899277871387at_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7094_sup__mono,axiom,
    ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C )
     => ( ( ord_less_eq_set_int @ B3 @ D )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ ( sup_sup_set_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7095_sup__mono,axiom,
    ! [A2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ C )
     => ( ( ord_less_eq_rat @ B3 @ D )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7096_sup__mono,axiom,
    ! [A2: nat,C: nat,B3: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B3 @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7097_sup__mono,axiom,
    ! [A2: int,C: int,B3: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ( ord_less_eq_int @ B3 @ D )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ ( sup_sup_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_7098_sup__least,axiom,
    ! [Y3: set_nat,X2: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X2 )
     => ( ( ord_less_eq_set_nat @ Z @ X2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7099_sup__least,axiom,
    ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ Y3 @ X2 )
     => ( ( ord_le3000389064537975527at_nat @ Z @ X2 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7100_sup__least,axiom,
    ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y3 @ X2 )
     => ( ( ord_le1268244103169919719at_nat @ Z @ X2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7101_sup__least,axiom,
    ! [Y3: set_int,X2: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X2 )
     => ( ( ord_less_eq_set_int @ Z @ X2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7102_sup__least,axiom,
    ! [Y3: rat,X2: rat,Z: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ( ord_less_eq_rat @ Z @ X2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7103_sup__least,axiom,
    ! [Y3: nat,X2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( ord_less_eq_nat @ Z @ X2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7104_sup__least,axiom,
    ! [Y3: int,X2: int,Z: int] :
      ( ( ord_less_eq_int @ Y3 @ X2 )
     => ( ( ord_less_eq_int @ Z @ X2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y3 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_7105_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X: set_nat,Y: set_nat] :
          ( ( sup_sup_set_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7106_le__iff__sup,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7107_le__iff__sup,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7108_le__iff__sup,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( sup_sup_set_int @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7109_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( sup_sup_rat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7110_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( sup_sup_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7111_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( sup_sup_int @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_7112_sup_OorderE,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( A2
        = ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7113_sup_OorderE,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
     => ( A2
        = ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7114_sup_OorderE,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
     => ( A2
        = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7115_sup_OorderE,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( A2
        = ( sup_sup_set_int @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7116_sup_OorderE,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( A2
        = ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7117_sup_OorderE,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7118_sup_OorderE,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( A2
        = ( sup_sup_int @ A2 @ B3 ) ) ) ).

% sup.orderE
thf(fact_7119_sup_OorderI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( A2
        = ( sup_sup_set_nat @ A2 @ B3 ) )
     => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7120_sup_OorderI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( A2
        = ( sup_su718114333110466843at_nat @ A2 @ B3 ) )
     => ( ord_le3000389064537975527at_nat @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7121_sup_OorderI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( A2
        = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) )
     => ( ord_le1268244103169919719at_nat @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7122_sup_OorderI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2
        = ( sup_sup_set_int @ A2 @ B3 ) )
     => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7123_sup_OorderI,axiom,
    ! [A2: rat,B3: rat] :
      ( ( A2
        = ( sup_sup_rat @ A2 @ B3 ) )
     => ( ord_less_eq_rat @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7124_sup_OorderI,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B3 ) )
     => ( ord_less_eq_nat @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7125_sup_OorderI,axiom,
    ! [A2: int,B3: int] :
      ( ( A2
        = ( sup_sup_int @ A2 @ B3 ) )
     => ( ord_less_eq_int @ B3 @ A2 ) ) ).

% sup.orderI
thf(fact_7126_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
      ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
              ( ( ord_less_eq_set_nat @ Y4 @ X5 )
             => ( ( ord_less_eq_set_nat @ Z4 @ X5 )
               => ( ord_less_eq_set_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_sup_set_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7127_sup__unique,axiom,
    ! [F: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
      ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat,Z4: set_Pr8693737435421807431at_nat] :
              ( ( ord_le3000389064537975527at_nat @ Y4 @ X5 )
             => ( ( ord_le3000389064537975527at_nat @ Z4 @ X5 )
               => ( ord_le3000389064537975527at_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7128_sup__unique,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,Z4: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ Y4 @ X5 )
             => ( ( ord_le1268244103169919719at_nat @ Z4 @ X5 )
               => ( ord_le1268244103169919719at_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7129_sup__unique,axiom,
    ! [F: set_int > set_int > set_int,X2: set_int,Y3: set_int] :
      ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: set_int,Y4: set_int,Z4: set_int] :
              ( ( ord_less_eq_set_int @ Y4 @ X5 )
             => ( ( ord_less_eq_set_int @ Z4 @ X5 )
               => ( ord_less_eq_set_int @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_sup_set_int @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7130_sup__unique,axiom,
    ! [F: rat > rat > rat,X2: rat,Y3: rat] :
      ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: rat,Y4: rat,Z4: rat] :
              ( ( ord_less_eq_rat @ Y4 @ X5 )
             => ( ( ord_less_eq_rat @ Z4 @ X5 )
               => ( ord_less_eq_rat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_sup_rat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7131_sup__unique,axiom,
    ! [F: nat > nat > nat,X2: nat,Y3: nat] :
      ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: nat,Y4: nat,Z4: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X5 )
             => ( ( ord_less_eq_nat @ Z4 @ X5 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_sup_nat @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7132_sup__unique,axiom,
    ! [F: int > int > int,X2: int,Y3: int] :
      ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: int,Y4: int,Z4: int] :
              ( ( ord_less_eq_int @ Y4 @ X5 )
             => ( ( ord_less_eq_int @ Z4 @ X5 )
               => ( ord_less_eq_int @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
         => ( ( sup_sup_int @ X2 @ Y3 )
            = ( F @ X2 @ Y3 ) ) ) ) ) ).

% sup_unique
thf(fact_7133_sup_Oabsorb1,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7134_sup_Oabsorb1,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
     => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7135_sup_Oabsorb1,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7136_sup_Oabsorb1,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( sup_sup_set_int @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7137_sup_Oabsorb1,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( sup_sup_rat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7138_sup_Oabsorb1,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7139_sup_Oabsorb1,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( sup_sup_int @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_7140_sup_Oabsorb2,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( sup_sup_set_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7141_sup_Oabsorb2,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ B3 )
     => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7142_sup_Oabsorb2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B3 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7143_sup_Oabsorb2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( sup_sup_set_int @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7144_sup_Oabsorb2,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A2 @ B3 )
     => ( ( sup_sup_rat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7145_sup_Oabsorb2,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( sup_sup_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7146_sup_Oabsorb2,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( sup_sup_int @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb2
thf(fact_7147_sup__absorb1,axiom,
    ! [Y3: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y3 @ X2 )
     => ( ( sup_sup_set_nat @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7148_sup__absorb1,axiom,
    ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ Y3 @ X2 )
     => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7149_sup__absorb1,axiom,
    ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y3 @ X2 )
     => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7150_sup__absorb1,axiom,
    ! [Y3: set_int,X2: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X2 )
     => ( ( sup_sup_set_int @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7151_sup__absorb1,axiom,
    ! [Y3: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X2 )
     => ( ( sup_sup_rat @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7152_sup__absorb1,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ( sup_sup_nat @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7153_sup__absorb1,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ Y3 @ X2 )
     => ( ( sup_sup_int @ X2 @ Y3 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_7154_sup__absorb2,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y3 )
     => ( ( sup_sup_set_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7155_sup__absorb2,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ X2 @ Y3 )
     => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7156_sup__absorb2,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X2 @ Y3 )
     => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7157_sup__absorb2,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X2 @ Y3 )
     => ( ( sup_sup_set_int @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7158_sup__absorb2,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ( sup_sup_rat @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7159_sup__absorb2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( sup_sup_nat @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7160_sup__absorb2,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ( sup_sup_int @ X2 @ Y3 )
        = Y3 ) ) ).

% sup_absorb2
thf(fact_7161_sup_OboundedE,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_nat @ B3 @ A2 )
         => ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7162_sup_OboundedE,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
         => ~ ( ord_le3000389064537975527at_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7163_sup_OboundedE,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
         => ~ ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7164_sup_OboundedE,axiom,
    ! [B3: set_int,C: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_int @ B3 @ A2 )
         => ~ ( ord_less_eq_set_int @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7165_sup_OboundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_rat @ B3 @ A2 )
         => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7166_sup_OboundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
         => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7167_sup_OboundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_int @ B3 @ A2 )
         => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_7168_sup_OboundedI,axiom,
    ! [B3: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ A2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7169_sup_OboundedI,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
     => ( ( ord_le3000389064537975527at_nat @ C @ A2 )
       => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7170_sup_OboundedI,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ C @ A2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7171_sup_OboundedI,axiom,
    ! [B3: set_int,A2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( ord_less_eq_set_int @ C @ A2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7172_sup_OboundedI,axiom,
    ! [B3: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A2 )
     => ( ( ord_less_eq_rat @ C @ A2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7173_sup_OboundedI,axiom,
    ! [B3: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7174_sup_OboundedI,axiom,
    ! [B3: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A2 )
     => ( ( ord_less_eq_int @ C @ A2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_7175_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( A4
          = ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7176_sup_Oorder__iff,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
          ( A4
          = ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7177_sup_Oorder__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
          ( A4
          = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7178_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( A4
          = ( sup_sup_set_int @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7179_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( A4
          = ( sup_sup_rat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7180_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7181_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( A4
          = ( sup_sup_int @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_7182_sup_Ocobounded1,axiom,
    ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7183_sup_Ocobounded1,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7184_sup_Ocobounded1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7185_sup_Ocobounded1,axiom,
    ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ A2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7186_sup_Ocobounded1,axiom,
    ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ A2 @ ( sup_sup_rat @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7187_sup_Ocobounded1,axiom,
    ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7188_sup_Ocobounded1,axiom,
    ! [A2: int,B3: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B3 ) ) ).

% sup.cobounded1
thf(fact_7189_sup_Ocobounded2,axiom,
    ! [B3: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7190_sup_Ocobounded2,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B3 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7191_sup_Ocobounded2,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B3 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7192_sup_Ocobounded2,axiom,
    ! [B3: set_int,A2: set_int] : ( ord_less_eq_set_int @ B3 @ ( sup_sup_set_int @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7193_sup_Ocobounded2,axiom,
    ! [B3: rat,A2: rat] : ( ord_less_eq_rat @ B3 @ ( sup_sup_rat @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7194_sup_Ocobounded2,axiom,
    ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7195_sup_Ocobounded2,axiom,
    ! [B3: int,A2: int] : ( ord_less_eq_int @ B3 @ ( sup_sup_int @ A2 @ B3 ) ) ).

% sup.cobounded2
thf(fact_7196_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( sup_sup_set_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7197_sup_Oabsorb__iff1,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7198_sup_Oabsorb__iff1,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7199_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( sup_sup_set_int @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7200_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( sup_sup_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7201_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7202_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( sup_sup_int @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_7203_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( sup_sup_set_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7204_sup_Oabsorb__iff2,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
          ( ( sup_su718114333110466843at_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7205_sup_Oabsorb__iff2,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7206_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( sup_sup_set_int @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7207_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( sup_sup_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7208_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7209_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( sup_sup_int @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_7210_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A2 )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7211_sup_OcoboundedI1,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C @ A2 )
     => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7212_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A2 )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7213_sup_OcoboundedI1,axiom,
    ! [C: set_int,A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ C @ A2 )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7214_sup_OcoboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ A2 )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7215_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7216_sup_OcoboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% sup.coboundedI1
thf(fact_7217_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B3: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B3 )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7218_sup_OcoboundedI2,axiom,
    ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ C @ B3 )
     => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7219_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ B3 )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7220_sup_OcoboundedI2,axiom,
    ! [C: set_int,B3: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ C @ B3 )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7221_sup_OcoboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C @ B3 )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7222_sup_OcoboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ B3 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7223_sup_OcoboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( ord_less_eq_int @ C @ B3 )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% sup.coboundedI2
thf(fact_7224_less__infI1,axiom,
    ! [A2: set_nat,X2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ X2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI1
thf(fact_7225_less__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ X2 )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI1
thf(fact_7226_less__infI1,axiom,
    ! [A2: real,X2: real,B3: real] :
      ( ( ord_less_real @ A2 @ X2 )
     => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ X2 ) ) ).

% less_infI1
thf(fact_7227_less__infI1,axiom,
    ! [A2: rat,X2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ X2 )
     => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI1
thf(fact_7228_less__infI1,axiom,
    ! [A2: nat,X2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ X2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI1
thf(fact_7229_less__infI1,axiom,
    ! [A2: int,X2: int,B3: int] :
      ( ( ord_less_int @ A2 @ X2 )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).

% less_infI1
thf(fact_7230_less__infI2,axiom,
    ! [B3: set_nat,X2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B3 @ X2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI2
thf(fact_7231_less__infI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B3 @ X2 )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI2
thf(fact_7232_less__infI2,axiom,
    ! [B3: real,X2: real,A2: real] :
      ( ( ord_less_real @ B3 @ X2 )
     => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ X2 ) ) ).

% less_infI2
thf(fact_7233_less__infI2,axiom,
    ! [B3: rat,X2: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ X2 )
     => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI2
thf(fact_7234_less__infI2,axiom,
    ! [B3: nat,X2: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ X2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).

% less_infI2
thf(fact_7235_less__infI2,axiom,
    ! [B3: int,X2: int,A2: int] :
      ( ( ord_less_int @ B3 @ X2 )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).

% less_infI2
thf(fact_7236_inf_Oabsorb3,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B3 )
     => ( ( inf_inf_set_nat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_7237_inf_Oabsorb3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_7238_inf_Oabsorb3,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( inf_inf_real @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_7239_inf_Oabsorb3,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( inf_inf_rat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_7240_inf_Oabsorb3,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( inf_inf_nat @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_7241_inf_Oabsorb3,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( inf_inf_int @ A2 @ B3 )
        = A2 ) ) ).

% inf.absorb3
thf(fact_7242_inf_Oabsorb4,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B3 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb4
thf(fact_7243_inf_Oabsorb4,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B3 @ A2 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb4
thf(fact_7244_inf_Oabsorb4,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( inf_inf_real @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb4
thf(fact_7245_inf_Oabsorb4,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( inf_inf_rat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb4
thf(fact_7246_inf_Oabsorb4,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( inf_inf_nat @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb4
thf(fact_7247_inf_Oabsorb4,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( inf_inf_int @ A2 @ B3 )
        = B3 ) ) ).

% inf.absorb4
thf(fact_7248_inf_Ostrict__boundedE,axiom,
    ! [A2: set_nat,B3: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
     => ~ ( ( ord_less_set_nat @ A2 @ B3 )
         => ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_7249_inf_Ostrict__boundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
     => ~ ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
         => ~ ( ord_le7866589430770878221at_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_7250_inf_Ostrict__boundedE,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_real @ A2 @ ( inf_inf_real @ B3 @ C ) )
     => ~ ( ( ord_less_real @ A2 @ B3 )
         => ~ ( ord_less_real @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_7251_inf_Ostrict__boundedE,axiom,
    ! [A2: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
     => ~ ( ( ord_less_rat @ A2 @ B3 )
         => ~ ( ord_less_rat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_7252_inf_Ostrict__boundedE,axiom,
    ! [A2: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
     => ~ ( ( ord_less_nat @ A2 @ B3 )
         => ~ ( ord_less_nat @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_7253_inf_Ostrict__boundedE,axiom,
    ! [A2: int,B3: int,C: int] :
      ( ( ord_less_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
     => ~ ( ( ord_less_int @ A2 @ B3 )
         => ~ ( ord_less_int @ A2 @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_7254_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( A4
            = ( inf_inf_set_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_7255_inf_Ostrict__order__iff,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( A4
            = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_7256_inf_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( A4
            = ( inf_inf_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_7257_inf_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( A4
            = ( inf_inf_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_7258_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( A4
            = ( inf_inf_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_7259_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( A4
            = ( inf_inf_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_7260_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_7261_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_7262_inf_Ostrict__coboundedI1,axiom,
    ! [A2: real,C: real,B3: real] :
      ( ( ord_less_real @ A2 @ C )
     => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_7263_inf_Ostrict__coboundedI1,axiom,
    ! [A2: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_7264_inf_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_7265_inf_Ostrict__coboundedI1,axiom,
    ! [A2: int,C: int,B3: int] :
      ( ( ord_less_int @ A2 @ C )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_7266_inf_Ostrict__coboundedI2,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B3 @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_7267_inf_Ostrict__coboundedI2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B3 @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_7268_inf_Ostrict__coboundedI2,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_real @ B3 @ C )
     => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_7269_inf_Ostrict__coboundedI2,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_7270_inf_Ostrict__coboundedI2,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_7271_inf_Ostrict__coboundedI2,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_int @ B3 @ C )
     => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_7272_less__supI1,axiom,
    ! [X2: set_nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ X2 @ A2 )
     => ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7273_less__supI1,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ X2 @ A2 )
     => ( ord_le6428140832669894131at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7274_less__supI1,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X2 @ A2 )
     => ( ord_le2604355607129572851at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7275_less__supI1,axiom,
    ! [X2: real,A2: real,B3: real] :
      ( ( ord_less_real @ X2 @ A2 )
     => ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7276_less__supI1,axiom,
    ! [X2: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ X2 @ A2 )
     => ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7277_less__supI1,axiom,
    ! [X2: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ X2 @ A2 )
     => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7278_less__supI1,axiom,
    ! [X2: int,A2: int,B3: int] :
      ( ( ord_less_int @ X2 @ A2 )
     => ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% less_supI1
thf(fact_7279_less__supI2,axiom,
    ! [X2: set_nat,B3: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ X2 @ B3 )
     => ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7280_less__supI2,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ X2 @ B3 )
     => ( ord_le6428140832669894131at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7281_less__supI2,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ X2 @ B3 )
     => ( ord_le2604355607129572851at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7282_less__supI2,axiom,
    ! [X2: real,B3: real,A2: real] :
      ( ( ord_less_real @ X2 @ B3 )
     => ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7283_less__supI2,axiom,
    ! [X2: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ X2 @ B3 )
     => ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7284_less__supI2,axiom,
    ! [X2: nat,B3: nat,A2: nat] :
      ( ( ord_less_nat @ X2 @ B3 )
     => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7285_less__supI2,axiom,
    ! [X2: int,B3: int,A2: int] :
      ( ( ord_less_int @ X2 @ B3 )
     => ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% less_supI2
thf(fact_7286_sup_Oabsorb3,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B3 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7287_sup_Oabsorb3,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ B3 @ A2 )
     => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7288_sup_Oabsorb3,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ B3 @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7289_sup_Oabsorb3,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( ( sup_sup_real @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7290_sup_Oabsorb3,axiom,
    ! [B3: rat,A2: rat] :
      ( ( ord_less_rat @ B3 @ A2 )
     => ( ( sup_sup_rat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7291_sup_Oabsorb3,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ord_less_nat @ B3 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7292_sup_Oabsorb3,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ B3 @ A2 )
     => ( ( sup_sup_int @ A2 @ B3 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_7293_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B3 )
     => ( ( sup_sup_set_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7294_sup_Oabsorb4,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A2 @ B3 )
     => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7295_sup_Oabsorb4,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A2 @ B3 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7296_sup_Oabsorb4,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( sup_sup_real @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7297_sup_Oabsorb4,axiom,
    ! [A2: rat,B3: rat] :
      ( ( ord_less_rat @ A2 @ B3 )
     => ( ( sup_sup_rat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7298_sup_Oabsorb4,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( sup_sup_nat @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7299_sup_Oabsorb4,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( sup_sup_int @ A2 @ B3 )
        = B3 ) ) ).

% sup.absorb4
thf(fact_7300_sup_Ostrict__boundedE,axiom,
    ! [B3: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_set_nat @ B3 @ A2 )
         => ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7301_sup_Ostrict__boundedE,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_le6428140832669894131at_nat @ B3 @ A2 )
         => ~ ( ord_le6428140832669894131at_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7302_sup_Ostrict__boundedE,axiom,
    ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_le2604355607129572851at_nat @ B3 @ A2 )
         => ~ ( ord_le2604355607129572851at_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7303_sup_Ostrict__boundedE,axiom,
    ! [B3: real,C: real,A2: real] :
      ( ( ord_less_real @ ( sup_sup_real @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_real @ B3 @ A2 )
         => ~ ( ord_less_real @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7304_sup_Ostrict__boundedE,axiom,
    ! [B3: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_rat @ B3 @ A2 )
         => ~ ( ord_less_rat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7305_sup_Ostrict__boundedE,axiom,
    ! [B3: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_nat @ B3 @ A2 )
         => ~ ( ord_less_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7306_sup_Ostrict__boundedE,axiom,
    ! [B3: int,C: int,A2: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
     => ~ ( ( ord_less_int @ B3 @ A2 )
         => ~ ( ord_less_int @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_7307_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( A4
            = ( sup_sup_set_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7308_sup_Ostrict__order__iff,axiom,
    ( ord_le6428140832669894131at_nat
    = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
          ( ( A4
            = ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7309_sup_Ostrict__order__iff,axiom,
    ( ord_le2604355607129572851at_nat
    = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
          ( ( A4
            = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7310_sup_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( A4
            = ( sup_sup_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7311_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( A4
            = ( sup_sup_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7312_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7313_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( A4
            = ( sup_sup_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_7314_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ C @ A2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7315_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ C @ A2 )
     => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7316_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C @ A2 )
     => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7317_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A2: real,B3: real] :
      ( ( ord_less_real @ C @ A2 )
     => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7318_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A2: rat,B3: rat] :
      ( ( ord_less_rat @ C @ A2 )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7319_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B3: nat] :
      ( ( ord_less_nat @ C @ A2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7320_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B3: int] :
      ( ( ord_less_int @ C @ A2 )
     => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_7321_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B3: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ C @ B3 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7322_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ C @ B3 )
     => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7323_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ C @ B3 )
     => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7324_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B3: real,A2: real] :
      ( ( ord_less_real @ C @ B3 )
     => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7325_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B3: rat,A2: rat] :
      ( ( ord_less_rat @ C @ B3 )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7326_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B3: nat,A2: nat] :
      ( ( ord_less_nat @ C @ B3 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7327_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B3: int,A2: int] :
      ( ( ord_less_int @ C @ B3 )
     => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_7328_distrib__sup__le,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7329_distrib__sup__le,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7330_distrib__sup__le,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X2 @ ( inf_in4302113700860409141at_nat @ Y3 @ Z ) ) @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) @ ( sup_su718114333110466843at_nat @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7331_distrib__sup__le,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ ( inf_in7913087082777306421at_nat @ Y3 @ Z ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7332_distrib__sup__le,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) ) @ ( inf_inf_set_int @ ( sup_sup_set_int @ X2 @ Y3 ) @ ( sup_sup_set_int @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7333_distrib__sup__le,axiom,
    ! [X2: rat,Y3: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X2 @ Y3 ) @ ( sup_sup_rat @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7334_distrib__sup__le,axiom,
    ! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7335_distrib__sup__le,axiom,
    ! [X2: int,Y3: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X2 @ Y3 ) @ ( sup_sup_int @ X2 @ Z ) ) ) ).

% distrib_sup_le
thf(fact_7336_distrib__inf__le,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7337_distrib__inf__le,axiom,
    ! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7338_distrib__inf__le,axiom,
    ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ Y3 ) @ ( inf_in4302113700860409141at_nat @ X2 @ Z ) ) @ ( inf_in4302113700860409141at_nat @ X2 @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7339_distrib__inf__le,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y3 ) @ ( inf_in7913087082777306421at_nat @ X2 @ Z ) ) @ ( inf_in7913087082777306421at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7340_distrib__inf__le,axiom,
    ! [X2: set_int,Y3: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ ( inf_inf_set_int @ X2 @ Z ) ) @ ( inf_inf_set_int @ X2 @ ( sup_sup_set_int @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7341_distrib__inf__le,axiom,
    ! [X2: rat,Y3: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ ( inf_inf_rat @ X2 @ Z ) ) @ ( inf_inf_rat @ X2 @ ( sup_sup_rat @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7342_distrib__inf__le,axiom,
    ! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7343_distrib__inf__le,axiom,
    ! [X2: int,Y3: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X2 @ Y3 ) @ ( inf_inf_int @ X2 @ Z ) ) @ ( inf_inf_int @ X2 @ ( sup_sup_int @ Y3 @ Z ) ) ) ).

% distrib_inf_le
thf(fact_7344_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K4: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K4 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7345_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K4: real,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7346_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ N @ A3 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).

% set_encode_insert
thf(fact_7347_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys2 ) @ N )
        = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7348_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) @ N )
        = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7349_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs @ Ys2 ) @ N )
        = ( product_Pair_int_o @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7350_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys2 ) @ N )
        = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7351_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7352_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7353_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7354_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7355_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys2 ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7356_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys2 ) @ N )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_7357_unset__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A2 )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7358_unset__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A2 )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7359_unset__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A2 )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_7360_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X2: 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 @ X2 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_7361_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X2: 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 @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_7362_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).

% set_vebt'_def
thf(fact_7363_finite__Collect__conjI,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ( finite_finite_real @ ( collect_real @ P ) )
        | ( finite_finite_real @ ( collect_real @ Q ) ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7364_finite__Collect__conjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7365_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7366_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7367_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7368_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7369_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7370_finite__Collect__conjI,axiom,
    ! [P: extended_enat > $o,Q: extended_enat > $o] :
      ( ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
        | ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) )
     => ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X: extended_enat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7371_finite__Collect__disjI,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_real @ ( collect_real @ P ) )
        & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7372_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7373_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7374_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7375_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7376_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7377_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7378_finite__Collect__disjI,axiom,
    ! [P: extended_enat > $o,Q: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X: extended_enat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
        & ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_7379_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T @ Y )
                & ( ord_less_nat @ X2 @ Y ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_7380_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T @ Y )
                & ( ord_less_nat @ Y @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_7381_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_7382_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_7383_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_7384_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7385_finite__Collect__subsets,axiom,
    ! [A3: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7386_finite__Collect__subsets,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7387_finite__Collect__subsets,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite5468666774076196335d_enat
        @ ( collec2260605976452661553d_enat
          @ ^ [B6: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7388_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7389_singleton__conv2,axiom,
    ! [A2: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y6: product_prod_nat_nat,Z3: product_prod_nat_nat] : Y6 = Z3
          @ A2 ) )
      = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_7390_singleton__conv2,axiom,
    ! [A2: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y6: list_nat,Z3: list_nat] : Y6 = Z3
          @ A2 ) )
      = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_7391_singleton__conv2,axiom,
    ! [A2: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y6: set_nat,Z3: set_nat] : Y6 = Z3
          @ A2 ) )
      = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_7392_singleton__conv2,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ( ^ [Y6: real,Z3: real] : Y6 = Z3
          @ A2 ) )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_7393_singleton__conv2,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ( ^ [Y6: $o,Z3: $o] : Y6 = Z3
          @ A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_7394_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z3: nat] : Y6 = Z3
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_7395_singleton__conv2,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ( ^ [Y6: int,Z3: int] : Y6 = Z3
          @ A2 ) )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_7396_singleton__conv,axiom,
    ! [A2: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : X = A2 )
      = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_7397_singleton__conv,axiom,
    ! [A2: list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : X = A2 )
      = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_7398_singleton__conv,axiom,
    ! [A2: set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : X = A2 )
      = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_7399_singleton__conv,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ^ [X: real] : X = A2 )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_7400_singleton__conv,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ^ [X: $o] : X = A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_7401_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : X = A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_7402_singleton__conv,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ^ [X: int] : X = A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_7403_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_7404_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_7405_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_7406_finite__interval__int1,axiom,
    ! [A2: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A2 @ I4 )
            & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int1
thf(fact_7407_finite__interval__int4,axiom,
    ! [A2: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A2 @ I4 )
            & ( ord_less_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int4
thf(fact_7408_norm__eq__zero,axiom,
    ! [X2: real] :
      ( ( ( real_V7735802525324610683m_real @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_7409_norm__eq__zero,axiom,
    ! [X2: complex] :
      ( ( ( real_V1022390504157884413omplex @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_7410_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_7411_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_7412_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_7413_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_7414_finite__interval__int2,axiom,
    ! [A2: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A2 @ I4 )
            & ( ord_less_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int2
thf(fact_7415_finite__interval__int3,axiom,
    ! [A2: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A2 @ I4 )
            & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int3
thf(fact_7416_zero__less__norm__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
      = ( X2 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_7417_zero__less__norm__iff,axiom,
    ! [X2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
      = ( X2 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_7418_norm__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_7419_norm__le__zero__iff,axiom,
    ! [X2: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
      = ( X2 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_7420_del__x__not__mia,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X2 = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_7421_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = 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_7422_del__x__not__mi,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_7423_del__in__range,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ 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 ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = 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 ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = 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 ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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 @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = 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 ) ) ) ) ) ) @ X2 ) @ ( 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_7424_del__x__mi,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 ) @ X2 )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_7425_del__x__mi__lets__in,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_7426_del__x__mi__lets__in__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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 @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
                            = ( 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_7427_del__x__mia,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ 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 ) @ X2 )
            = ( 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_7428_pred__less__length__list,axiom,
    ! [Deg: nat,X2: 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 @ X2 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_7429_pred__lesseq__max,axiom,
    ! [Deg: nat,X2: 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 @ X2 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_7430_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7431_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7432_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7433_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7434_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).

% max_def_raw
thf(fact_7435_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_7436_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z2: complex] :
                  ( ( power_power_complex @ Z2 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_7437_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_7438_minus__set__def,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( minus_minus_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 )
            @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_7439_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X: real] : ( member_real @ X @ A6 )
            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_7440_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_7441_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_7442_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X: int] : ( member_int @ X @ A6 )
            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_7443_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 )
            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_7444_set__diff__eq,axiom,
    ( minus_minus_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A6 )
              & ~ ( member_o @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_7445_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A6 )
              & ~ ( member_real @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_7446_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A6 )
              & ~ ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_7447_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A6 )
              & ~ ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_7448_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A6 )
              & ~ ( member_int @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_7449_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_7450_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_7451_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_7452_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_7453_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_7454_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_7455_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_7456_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( ord_less_o_o
          @ ^ [X: $o] : ( member_o @ X @ A6 )
          @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_7457_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
          @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_7458_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_7459_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_7460_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A6 )
              | ( member_o @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7461_Un__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A6 )
              | ( member_real @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7462_Un__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A6 )
              | ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7463_Un__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A6 )
              | ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7464_Un__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A6 )
              | ( member_int @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7465_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              | ( member_nat @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7466_Un__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ A6 )
              | ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7467_Un__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X @ A6 )
              | ( member8757157785044589968at_nat @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_7468_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 )
            @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7469_sup__set__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( sup_sup_real_o
            @ ^ [X: real] : ( member_real @ X @ A6 )
            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7470_sup__set__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( sup_sup_list_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7471_sup__set__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( sup_sup_set_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7472_sup__set__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( sup_sup_int_o
            @ ^ [X: int] : ( member_int @ X @ A6 )
            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7473_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 )
            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7474_sup__set__def,axiom,
    ( sup_su718114333110466843at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( sup_su8986005896011022210_nat_o
            @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A6 )
            @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7475_sup__set__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ( sup_su2080679670758317954_nat_o
            @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A6 )
            @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_7476_Collect__disj__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7477_Collect__disj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7478_Collect__disj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7479_Collect__disj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7480_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7481_Collect__disj__eq,axiom,
    ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su718114333110466843at_nat @ ( collec7088162979684241874at_nat @ P ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7482_Collect__disj__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_7483_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A4: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7484_insert__def,axiom,
    ( insert_o
    = ( ^ [A4: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X: $o] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7485_insert__def,axiom,
    ( insert_real
    = ( ^ [A4: real] :
          ( sup_sup_set_real
          @ ( collect_real
            @ ^ [X: real] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7486_insert__def,axiom,
    ( insert_list_nat
    = ( ^ [A4: list_nat] :
          ( sup_sup_set_list_nat
          @ ( collect_list_nat
            @ ^ [X: list_nat] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7487_insert__def,axiom,
    ( insert_set_nat
    = ( ^ [A4: set_nat] :
          ( sup_sup_set_set_nat
          @ ( collect_set_nat
            @ ^ [X: set_nat] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7488_insert__def,axiom,
    ( insert_int
    = ( ^ [A4: int] :
          ( sup_sup_set_int
          @ ( collect_int
            @ ^ [X: int] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7489_insert__def,axiom,
    ( insert_nat
    = ( ^ [A4: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7490_insert__def,axiom,
    ( insert5050368324300391991at_nat
    = ( ^ [A4: produc859450856879609959at_nat] :
          ( sup_su718114333110466843at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7491_insert__def,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A4: produc3843707927480180839at_nat] :
          ( sup_su5525570899277871387at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] : X = A4 ) ) ) ) ).

% insert_def
thf(fact_7492_Compl__eq,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ~ ( member_o @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7493_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ~ ( member_real @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7494_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ~ ( member_list_nat @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7495_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ~ ( member_set_nat @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7496_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ~ ( member_nat @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7497_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ~ ( member_int @ X @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7498_Collect__neg__eq,axiom,
    ! [P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ~ ( P @ X ) )
      = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7499_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ~ ( P @ X ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7500_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ~ ( P @ X ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7501_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ~ ( P @ X ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7502_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ~ ( P @ X ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7503_uminus__set__def,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A6: set_o] :
          ( collect_o
          @ ( uminus_uminus_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7504_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X: real] : ( member_real @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7505_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7506_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7507_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7508_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X: int] : ( member_int @ X @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7509_not__finite__existsD,axiom,
    ! [P: real > $o] :
      ( ~ ( finite_finite_real @ ( collect_real @ P ) )
     => ? [X_1: real] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7510_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_1: list_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7511_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7512_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7513_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7514_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_1: complex] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7515_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7516_not__finite__existsD,axiom,
    ! [P: extended_enat > $o] :
      ( ~ ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
     => ? [X_1: extended_enat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_7517_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B2: set_nat,R: $o > nat > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7518_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_nat,R: real > nat > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7519_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B2: set_int,R: $o > int > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A3 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: int] :
              ( ( member_int @ X5 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7520_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_int,R: real > int > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A3 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: int] :
              ( ( member_int @ X5 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7521_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B2: set_complex,R: $o > complex > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A3 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: complex] :
              ( ( member_complex @ X5 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7522_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_complex,R: real > complex > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A3 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: complex] :
              ( ( member_complex @ X5 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7523_pigeonhole__infinite__rel,axiom,
    ! [A3: set_o,B2: set_Extended_enat,R: $o > extended_enat > $o] :
      ( ~ ( finite_finite_o @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A3 )
             => ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ B2 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A4: $o] :
                        ( ( member_o @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7524_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B2: set_Extended_enat,R: real > extended_enat > $o] :
      ( ~ ( finite_finite_real @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A3 )
             => ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A4: real] :
                        ( ( member_real @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7525_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7526_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X5 @ Xa ) ) )
         => ? [X5: int] :
              ( ( member_int @ X5 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A3 )
                        & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_7527_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
      ( ( ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_7528_Collect__conv__if,axiom,
    ! [P: list_nat > $o,A2: list_nat] :
      ( ( ( P @ A2 )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_7529_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ( ( P @ A2 )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_7530_Collect__conv__if,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_7531_Collect__conv__if,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_7532_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_7533_Collect__conv__if,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_7534_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
      ( ( ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7535_Collect__conv__if2,axiom,
    ! [P: list_nat > $o,A2: list_nat] :
      ( ( ( P @ A2 )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7536_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ( ( P @ A2 )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7537_Collect__conv__if2,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_7538_Collect__conv__if2,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_7539_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_7540_Collect__conv__if2,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_7541_insert__Collect,axiom,
    ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U2: product_prod_nat_nat] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7542_insert__Collect,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( insert_o @ A2 @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7543_insert__Collect,axiom,
    ! [A2: real,P: real > $o] :
      ( ( insert_real @ A2 @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7544_insert__Collect,axiom,
    ! [A2: list_nat,P: list_nat > $o] :
      ( ( insert_list_nat @ A2 @ ( collect_list_nat @ P ) )
      = ( collect_list_nat
        @ ^ [U2: list_nat] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7545_insert__Collect,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( insert_set_nat @ A2 @ ( collect_set_nat @ P ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7546_insert__Collect,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7547_insert__Collect,axiom,
    ! [A2: int,P: int > $o] :
      ( ( insert_int @ A2 @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_7548_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( X = A4 )
              | ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7549_insert__compr,axiom,
    ( insert_o
    = ( ^ [A4: $o,B6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( X = A4 )
              | ( member_o @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7550_insert__compr,axiom,
    ( insert_real
    = ( ^ [A4: real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( X = A4 )
              | ( member_real @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7551_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A4: list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( X = A4 )
              | ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7552_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A4: set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( X = A4 )
              | ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7553_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A4 )
              | ( member_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7554_insert__compr,axiom,
    ( insert_int
    = ( ^ [A4: int,B6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( X = A4 )
              | ( member_int @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_7555_empty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X: list_nat] : $false ) ) ).

% empty_def
thf(fact_7556_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X: set_nat] : $false ) ) ).

% empty_def
thf(fact_7557_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X: real] : $false ) ) ).

% empty_def
thf(fact_7558_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X: $o] : $false ) ) ).

% empty_def
thf(fact_7559_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_7560_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X: int] : $false ) ) ).

% empty_def
thf(fact_7561_Collect__imp__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_real @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) @ ( collect_real @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7562_Collect__imp__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7563_Collect__imp__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7564_Collect__imp__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7565_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7566_Collect__imp__eq,axiom,
    ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ ( collec7088162979684241874at_nat @ P ) ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7567_Collect__imp__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P ) ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_7568_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_7569_bot__empty__eq2,axiom,
    ( bot_bo5043116465536727218_nat_o
    = ( ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ bot_bo232370072503712749on_nat ) ) ) ).

% bot_empty_eq2
thf(fact_7570_bot__empty__eq2,axiom,
    ( bot_bo3364206721330744218_nat_o
    = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ bot_bo4948859079157340979at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_7571_bot__empty__eq2,axiom,
    ( bot_bo394778441745866138_nat_o
    = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_7572_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_7573_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_7574_pred__subset__eq,axiom,
    ! [R: set_real,S: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X: real] : ( member_real @ X @ R )
        @ ^ [X: real] : ( member_real @ X @ S ) )
      = ( ord_less_eq_set_real @ R @ S ) ) ).

% pred_subset_eq
thf(fact_7575_pred__subset__eq,axiom,
    ! [R: set_o,S: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X: $o] : ( member_o @ X @ R )
        @ ^ [X: $o] : ( member_o @ X @ S ) )
      = ( ord_less_eq_set_o @ R @ S ) ) ).

% pred_subset_eq
thf(fact_7576_pred__subset__eq,axiom,
    ! [R: set_set_nat,S: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
        @ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
      = ( ord_le6893508408891458716et_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_7577_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_7578_pred__subset__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X: int] : ( member_int @ X @ R )
        @ ^ [X: int] : ( member_int @ X @ S ) )
      = ( ord_less_eq_set_int @ R @ S ) ) ).

% pred_subset_eq
thf(fact_7579_prop__restrict,axiom,
    ! [X2: $o,Z6: set_o,X6: set_o,P: $o > $o] :
      ( ( member_o @ X2 @ Z6 )
     => ( ( ord_less_eq_set_o @ Z6
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7580_prop__restrict,axiom,
    ! [X2: real,Z6: set_real,X6: set_real,P: real > $o] :
      ( ( member_real @ X2 @ Z6 )
     => ( ( ord_less_eq_set_real @ Z6
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7581_prop__restrict,axiom,
    ! [X2: list_nat,Z6: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X2 @ Z6 )
     => ( ( ord_le6045566169113846134st_nat @ Z6
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( member_list_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7582_prop__restrict,axiom,
    ! [X2: set_nat,Z6: set_set_nat,X6: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X2 @ Z6 )
     => ( ( ord_le6893508408891458716et_nat @ Z6
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7583_prop__restrict,axiom,
    ! [X2: nat,Z6: set_nat,X6: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ Z6 )
     => ( ( ord_less_eq_set_nat @ Z6
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7584_prop__restrict,axiom,
    ! [X2: int,Z6: set_int,X6: set_int,P: int > $o] :
      ( ( member_int @ X2 @ Z6 )
     => ( ( ord_less_eq_set_int @ Z6
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ X6 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_7585_Collect__restrict,axiom,
    ! [X6: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_7586_Collect__restrict,axiom,
    ! [X6: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_7587_Collect__restrict,axiom,
    ! [X6: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_7588_Collect__restrict,axiom,
    ! [X6: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_7589_Collect__restrict,axiom,
    ! [X6: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_7590_Collect__restrict,axiom,
    ! [X6: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ X6 )
            & ( P @ X ) ) )
      @ X6 ) ).

% Collect_restrict
thf(fact_7591_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_7592_less__eq__set__def,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( ord_less_eq_o_o
          @ ^ [X: $o] : ( member_o @ X @ A6 )
          @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_7593_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
          @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_7594_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_7595_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_7596_Collect__subset,axiom,
    ! [A3: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_7597_Collect__subset,axiom,
    ! [A3: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_7598_Collect__subset,axiom,
    ! [A3: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_7599_Collect__subset,axiom,
    ! [A3: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_7600_Collect__subset,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_7601_Collect__subset,axiom,
    ! [A3: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_7602_pred__subset__eq2,axiom,
    ! [R: set_Pr6588086440996610945on_nat,S: set_Pr6588086440996610945on_nat] :
      ( ( ord_le8905833333647802342_nat_o
        @ ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ R )
        @ ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ S ) )
      = ( ord_le6406482658798684961on_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_7603_pred__subset__eq2,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
        @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S ) )
      = ( ord_le5997549366648089703at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_7604_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
        @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S ) )
      = ( ord_le1268244103169919719at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_7605_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_7606_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) )
      = ( ord_le2843351958646193337nt_int @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_7607_Int__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A6 )
              & ( member_o @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7608_Int__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A6 )
              & ( member_real @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7609_Int__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A6 )
              & ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7610_Int__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A6 )
              & ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7611_Int__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A6 )
              & ( member_int @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7612_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              & ( member_nat @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7613_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A6 )
              & ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_7614_Int__Collect,axiom,
    ! [X2: $o,A3: set_o,P: $o > $o] :
      ( ( member_o @ X2 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
      = ( ( member_o @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7615_Int__Collect,axiom,
    ! [X2: real,A3: set_real,P: real > $o] :
      ( ( member_real @ X2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) )
      = ( ( member_real @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7616_Int__Collect,axiom,
    ! [X2: list_nat,A3: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X2 @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) )
      = ( ( member_list_nat @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7617_Int__Collect,axiom,
    ! [X2: set_nat,A3: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) )
      = ( ( member_set_nat @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7618_Int__Collect,axiom,
    ! [X2: int,A3: set_int,P: int > $o] :
      ( ( member_int @ X2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) )
      = ( ( member_int @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7619_Int__Collect,axiom,
    ! [X2: nat,A3: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7620_Int__Collect,axiom,
    ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) )
      = ( ( member8440522571783428010at_nat @ X2 @ A3 )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_7621_inf__set__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A6: set_o,B6: set_o] :
          ( collect_o
          @ ( inf_inf_o_o
            @ ^ [X: $o] : ( member_o @ X @ A6 )
            @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7622_inf__set__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( inf_inf_real_o
            @ ^ [X: real] : ( member_real @ X @ A6 )
            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7623_inf__set__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( inf_inf_list_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7624_inf__set__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( inf_inf_set_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7625_inf__set__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( inf_inf_int_o
            @ ^ [X: int] : ( member_int @ X @ A6 )
            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7626_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 )
            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7627_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_7628_Collect__conj__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_7629_Collect__conj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_7630_Collect__conj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_7631_Collect__conj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_7632_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_7633_Collect__conj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_7634_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).

% set_vebt_def
thf(fact_7635_finite__int__segment,axiom,
    ! [A2: real,B3: real] :
      ( finite_finite_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ring_1_Ints_real )
            & ( ord_less_eq_real @ A2 @ X )
            & ( ord_less_eq_real @ X @ B3 ) ) ) ) ).

% finite_int_segment
thf(fact_7636_finite__int__segment,axiom,
    ! [A2: rat,B3: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [X: rat] :
            ( ( member_rat @ X @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ A2 @ X )
            & ( ord_less_eq_rat @ X @ B3 ) ) ) ) ).

% finite_int_segment
thf(fact_7637_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_7638_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_7639_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_7640_finite__abs__int__segment,axiom,
    ! [A2: real] :
      ( finite_finite_real
      @ ( collect_real
        @ ^ [K3: real] :
            ( ( member_real @ K3 @ ring_1_Ints_real )
            & ( ord_less_eq_real @ ( abs_abs_real @ K3 ) @ A2 ) ) ) ) ).

% finite_abs_int_segment
thf(fact_7641_finite__abs__int__segment,axiom,
    ! [A2: rat] :
      ( finite_finite_rat
      @ ( collect_rat
        @ ^ [K3: rat] :
            ( ( member_rat @ K3 @ ring_1_Ints_rat )
            & ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A2 ) ) ) ) ).

% finite_abs_int_segment
thf(fact_7642_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_7643_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M5 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_7644_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M5 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_7645_norm__not__less__zero,axiom,
    ! [X2: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_7646_norm__ge__zero,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% norm_ge_zero
thf(fact_7647_complex__mod__minus__le__complex__mod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_7648_complex__mod__triangle__ineq2,axiom,
    ! [B3: complex,A2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B3 @ A2 ) ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ A2 ) ) ).

% complex_mod_triangle_ineq2
thf(fact_7649_set__encode__eq,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ( nat_set_encode @ A3 )
            = ( nat_set_encode @ B2 ) )
          = ( A3 = B2 ) ) ) ) ).

% set_encode_eq
thf(fact_7650_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z2: real] :
              ( ( power_power_real @ Z2 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_7651_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_7652_card__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ord_less_eq_nat
        @ ( finite_card_real
          @ ( collect_real
            @ ^ [Z2: real] :
                ( ( power_power_real @ Z2 @ N )
                = one_one_real ) ) )
        @ N ) ) ).

% card_roots_unity
thf(fact_7653_card__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ord_less_eq_nat
        @ ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        @ N ) ) ).

% card_roots_unity
thf(fact_7654_finite__lists__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7655_finite__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
              & ( ( size_s5460976970255530739at_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7656_finite__lists__length__eq,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
              & ( ( size_s3941691890525107288d_enat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7657_finite__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7658_finite__lists__length__eq,axiom,
    ! [A3: set_o,N: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
              & ( ( size_size_list_o @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7659_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
              & ( ( size_size_list_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7660_finite__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
              & ( ( size_size_list_int @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7661_card__lists__length__eq,axiom,
    ! [A3: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A3 )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs2: list_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A3 )
                & ( ( size_s3023201423986296836st_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_list_nat @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7662_card__lists__length__eq,axiom,
    ! [A3: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs2: list_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A3 )
                & ( ( size_s3254054031482475050et_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_set_nat @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7663_card__lists__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs2: list_complex] :
                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
                & ( ( size_s3451745648224563538omplex @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_complex @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7664_card__lists__length__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite249151656366948015at_nat
          @ ( collec3343600615725829874at_nat
            @ ^ [Xs2: list_P6011104703257516679at_nat] :
                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
                & ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite711546835091564841at_nat @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7665_card__lists__length__eq,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite7441382602597825044d_enat
          @ ( collec8433460942617342167d_enat
            @ ^ [Xs2: list_Extended_enat] :
                ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
                & ( ( size_s3941691890525107288d_enat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite121521170596916366d_enat @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7666_card__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs2: list_VEBT_VEBT] :
                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7667_card__lists__length__eq,axiom,
    ! [A3: set_o,N: nat] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_card_list_o
          @ ( collect_list_o
            @ ^ [Xs2: list_o] :
                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
                & ( ( size_size_list_o @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_o @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7668_card__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [Xs2: list_nat] :
                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
                & ( ( size_size_list_nat @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_nat @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7669_card__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_card_list_int
          @ ( collect_list_int
            @ ^ [Xs2: list_int] :
                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
                & ( ( size_size_list_int @ Xs2 )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_int @ A3 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_7670_finite__lists__length__le,axiom,
    ! [A3: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7671_finite__lists__length__le,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7672_finite__lists__length__le,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7673_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A3 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7674_finite__lists__length__le,axiom,
    ! [A3: set_o,N: nat] :
      ( ( finite_finite_o @ A3 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7675_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7676_finite__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( ( finite_finite_int @ A3 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7677_nonzero__norm__divide,axiom,
    ! [B3: real,A2: real] :
      ( ( B3 != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B3 ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7678_nonzero__norm__divide,axiom,
    ! [B3: complex,A2: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7679_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z: real] :
      ( ( ( power_power_real @ W2 @ N )
        = ( power_power_real @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7680_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z: complex] :
      ( ( ( power_power_complex @ W2 @ N )
        = ( power_power_complex @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7681_norm__mult__less,axiom,
    ! [X2: real,R2: real,Y3: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_7682_norm__mult__less,axiom,
    ! [X2: complex,R2: real,Y3: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_7683_norm__mult__ineq,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_mult_ineq
thf(fact_7684_norm__mult__ineq,axiom,
    ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_mult_ineq
thf(fact_7685_norm__add__less,axiom,
    ! [X2: real,R2: real,Y3: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_add_less
thf(fact_7686_norm__add__less,axiom,
    ! [X2: complex,R2: real,Y3: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_add_less
thf(fact_7687_norm__triangle__lt,axiom,
    ! [X2: real,Y3: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_7688_norm__triangle__lt,axiom,
    ! [X2: complex,Y3: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_7689_norm__power__ineq,axiom,
    ! [X2: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).

% norm_power_ineq
thf(fact_7690_norm__power__ineq,axiom,
    ! [X2: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).

% norm_power_ineq
thf(fact_7691_norm__add__leD,axiom,
    ! [A2: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7692_norm__add__leD,axiom,
    ! [A2: complex,B3: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7693_norm__triangle__le,axiom,
    ! [X2: real,Y3: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_7694_norm__triangle__le,axiom,
    ! [X2: complex,Y3: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_7695_norm__triangle__ineq,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_7696_norm__triangle__ineq,axiom,
    ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_7697_norm__triangle__mono,axiom,
    ! [A2: real,R2: real,B3: real,S2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A2 ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ S2 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_triangle_mono
thf(fact_7698_norm__triangle__mono,axiom,
    ! [A2: complex,R2: real,B3: complex,S2: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A2 ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ S2 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_triangle_mono
thf(fact_7699_norm__diff__triangle__less,axiom,
    ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7700_norm__diff__triangle__less,axiom,
    ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7701_norm__triangle__sub,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) ) ) ).

% norm_triangle_sub
thf(fact_7702_norm__triangle__sub,axiom,
    ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) ) ) ).

% norm_triangle_sub
thf(fact_7703_norm__triangle__ineq4,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ).

% norm_triangle_ineq4
thf(fact_7704_norm__triangle__ineq4,axiom,
    ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ).

% norm_triangle_ineq4
thf(fact_7705_norm__diff__triangle__le,axiom,
    ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7706_norm__diff__triangle__le,axiom,
    ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7707_norm__triangle__le__diff,axiom,
    ! [X2: real,Y3: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_le_diff
thf(fact_7708_norm__triangle__le__diff,axiom,
    ! [X2: complex,Y3: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_le_diff
thf(fact_7709_norm__diff__ineq,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ).

% norm_diff_ineq
thf(fact_7710_norm__diff__ineq,axiom,
    ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ).

% norm_diff_ineq
thf(fact_7711_norm__triangle__ineq2,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).

% norm_triangle_ineq2
thf(fact_7712_norm__triangle__ineq2,axiom,
    ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).

% norm_triangle_ineq2
thf(fact_7713_norm__exp,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X2 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).

% norm_exp
thf(fact_7714_norm__exp,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X2 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).

% norm_exp
thf(fact_7715_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( nat_set_encode @ A3 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_7716_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_7717_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_7718_norm__diff__triangle__ineq,axiom,
    ! [A2: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7719_norm__diff__triangle__ineq,axiom,
    ! [A2: complex,B3: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B3 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7720_norm__sgn,axiom,
    ! [X2: real] :
      ( ( ( X2 = zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X2 ) )
          = zero_zero_real ) )
      & ( ( X2 != zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X2 ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_7721_norm__sgn,axiom,
    ! [X2: complex] :
      ( ( ( X2 = zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X2 ) )
          = zero_zero_real ) )
      & ( ( X2 != zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X2 ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_7722_norm__triangle__ineq3,axiom,
    ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).

% norm_triangle_ineq3
thf(fact_7723_norm__triangle__ineq3,axiom,
    ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).

% norm_triangle_ineq3
thf(fact_7724_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S2 ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_7725_norm__power__diff,axiom,
    ! [Z: real,W2: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ 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 @ Z @ M ) @ ( power_power_real @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W2 ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_7726_norm__power__diff,axiom,
    ! [Z: complex,W2: complex,M: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ 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 @ Z @ M ) @ ( power_power_complex @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W2 ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_7727_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_7728_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
      = ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ ( ord_less_nat @ X2 @ Mi )
            & ( ~ ( ord_less_nat @ X2 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X2 )
                & ( ~ ( ord_less_nat @ Ma @ X2 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_7729_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_7730_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7731_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7732_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( Y3
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y3 )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
               => ( Y3
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_7733_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_7734_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_7735_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_7736_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_7737_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_7738_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y3 )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y3 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y3
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
                 => ( Y3
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                   => ( Y3
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_7739_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_7740_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( Y3
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y3 )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y3 )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y3 )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                   => ( Y3
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_7741_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_7742_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7743_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7744_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y3
                    = ( vEBT_Leaf @ $true @ B ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y3
                        = ( vEBT_Leaf @ A @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y3
                        = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
             => ( Y3
               != ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
               => ( Y3
                 != ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
                 => ( Y3
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                   => ( Y3
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ 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 @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_7745_vebt__succ_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_7746_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_7747_vebt__delete_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X2 @ Mi )
          | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X2 @ Mi )
            | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_7748_vebt__delete_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
        = Y3 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y3
               != ( vEBT_Leaf @ $false @ B ) ) ) )
       => ( ! [A: $o] :
              ( ? [B: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ B ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y3
                 != ( vEBT_Leaf @ A @ $false ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ? [N3: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N3 ) ) )
                 => ( Y3
                   != ( vEBT_Leaf @ A @ B ) ) ) )
           => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                 => ( Y3
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y3
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y3
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y3
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y3
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y3
                                    = ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va2 ) )
                                          @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va2 ) )
                                          @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_7749_vebt__pred_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y3 != none_nat ) ) )
       => ( ! [A: $o] :
              ( ? [Uw2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A
                     => ( Y3
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A
                     => ( Y3 = none_nat ) ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ? [Va2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va2 ) ) )
                 => ~ ( ( B
                       => ( Y3
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B
                       => ( ( A
                           => ( Y3
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A
                           => ( Y3 = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y3 != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y3 != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y3
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y3
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_7750_vebt__succ_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
        = Y3 )
     => ( ! [Uu2: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ Uu2 @ B ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B
                   => ( Y3
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B
                   => ( Y3 = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N3: nat] :
                  ( Xa2
                  = ( suc @ N3 ) )
             => ( Y3 != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y3 != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y3 != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y3
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y3
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_7751_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7752_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7753_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7754_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7755_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7756_vebt__succ_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ B ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B
                     => ( Y3
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B
                     => ( Y3 = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N3: nat] :
                    ( ( Xa2
                      = ( suc @ N3 ) )
                   => ( ( Y3 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3 = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y3 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y3 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y3
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y3
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_7757_vebt__pred_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A
                       => ( Y3
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A
                       => ( Y3 = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [Va2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va2 ) ) )
                     => ( ( ( B
                           => ( Y3
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B
                           => ( ( A
                               => ( Y3
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A
                               => ( Y3 = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y3 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y3 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y3 = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y3
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y3
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_7758_vebt__delete_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y3
                    = ( vEBT_Leaf @ $false @ B ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y3
                      = ( vEBT_Leaf @ A @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [N3: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N3 ) ) )
                     => ( ( Y3
                          = ( vEBT_Leaf @ A @ B ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                   => ( ( Y3
                        = ( 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,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y3
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y3
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y3
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y3
                                      = ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va2 ) )
                                            @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va2 ) )
                                            @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_7759_monoseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7760_vebt__insert_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y3
                      = ( vEBT_Leaf @ $true @ B ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y3
                          = ( vEBT_Leaf @ A @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y3
                          = ( vEBT_Leaf @ A @ B ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y3
                    = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ Xa2 ) ) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y3
                      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
                   => ( ( Y3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                     => ( ( Y3
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ 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 @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_7761_monoseq__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).

% monoseq_realpow
thf(fact_7762_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y3
                  = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y3
                        = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
                    & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y3
                        = ( 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_7763_vebt__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y3
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y3
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y3
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                     => ( ( Y3
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_7764_vebt__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_7765_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_7766_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_7767_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y3
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
                 => ( ( Y3
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_7768_vebt__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_7769_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_7770_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y3
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y3
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
                   => ( ( Y3
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                     => ( ( Y3
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_7771_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_7772_ln__series,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X2 )
          = ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7773_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7774_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7775_arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arctan @ X2 )
        = ( 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 @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7776_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_7777_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7778_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7779_signed__take__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A2 )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7780_signed__take__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A2 )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7781_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_7782_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_7783_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_7784_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_7785_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_7786_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_7787_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_7788_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_7789_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_7790_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_7791_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7792_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7793_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N2: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7794_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N2: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7795_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N2: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7796_round__unique,axiom,
    ! [X2: real,Y3: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X2 )
          = Y3 ) ) ) ).

% round_unique
thf(fact_7797_round__unique,axiom,
    ! [X2: rat,Y3: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X2 )
          = Y3 ) ) ) ).

% round_unique
thf(fact_7798_summable__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ 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 @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7799_tanh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_7800_round__unique_H,axiom,
    ! [X2: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X2 )
        = N ) ) ).

% round_unique'
thf(fact_7801_round__unique_H,axiom,
    ! [X2: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X2 )
        = N ) ) ).

% round_unique'
thf(fact_7802_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_7803_tanh__real__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_real @ X2 @ Y3 ) ) ).

% tanh_real_less_iff
thf(fact_7804_tanh__real__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% tanh_real_le_iff
thf(fact_7805_summable__zero,axiom,
    ( summable_real
    @ ^ [N2: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7806_summable__zero,axiom,
    ( summable_nat
    @ ^ [N2: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7807_summable__zero,axiom,
    ( summable_int
    @ ^ [N2: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7808_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7809_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7810_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7811_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_7812_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_7813_tanh__real__neg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( tanh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_7814_tanh__real__pos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% tanh_real_pos_iff
thf(fact_7815_tanh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% tanh_real_nonneg_iff
thf(fact_7816_tanh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_7817_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7818_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7819_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7820_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7821_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_7822_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_7823_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_7824_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_7825_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_7826_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_7827_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_7828_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_7829_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7830_summable__comparison__test_H,axiom,
    ! [G3: nat > real,N6: nat,F: nat > real] :
      ( ( summable_real @ G3 )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7831_summable__comparison__test_H,axiom,
    ! [G3: nat > real,N6: nat,F: nat > complex] :
      ( ( summable_real @ G3 )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7832_summable__comparison__test,axiom,
    ! [F: nat > real,G3: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
     => ( ( summable_real @ G3 )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7833_summable__comparison__test,axiom,
    ! [F: nat > complex,G3: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
     => ( ( summable_real @ G3 )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7834_suminf__le,axiom,
    ! [F: nat > real,G3: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G3 @ N3 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G3 )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G3 ) ) ) ) ) ).

% suminf_le
thf(fact_7835_suminf__le,axiom,
    ! [F: nat > nat,G3: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G3 @ N3 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G3 )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G3 ) ) ) ) ) ).

% suminf_le
thf(fact_7836_suminf__le,axiom,
    ! [F: nat > int,G3: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G3 @ N3 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G3 )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G3 ) ) ) ) ) ).

% suminf_le
thf(fact_7837_summable__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_7838_summable__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_7839_summable__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_7840_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7841_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7842_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7843_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7844_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7845_powser__insidea,axiom,
    ! [F: nat > real,X2: real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
       => ( summable_real
          @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7846_powser__insidea,axiom,
    ! [F: nat > complex,X2: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
       => ( summable_real
          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7847_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7848_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7849_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7850_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N2: nat] :
                ( ( F @ N2 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7851_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N2: nat] :
                ( ( F @ N2 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7852_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N2: nat] :
                ( ( F @ N2 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7853_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7854_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7855_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7856_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).

% summable_zero_power'
thf(fact_7857_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).

% summable_zero_power'
thf(fact_7858_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).

% summable_zero_power'
thf(fact_7859_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).

% summable_0_powser
thf(fact_7860_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).

% summable_0_powser
thf(fact_7861_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G3: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
     => ( ( summable_real @ G3 )
       => ( summable_real
          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7862_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G3: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
     => ( ( summable_real @ G3 )
       => ( summable_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7863_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs
thf(fact_7864_tanh__real__lt__1,axiom,
    ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_7865_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7866_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7867_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7868_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7869_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7870_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7871_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_7872_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_7873_complete__algebra__summable__geometric,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X2 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7874_complete__algebra__summable__geometric,axiom,
    ! [X2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7875_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7876_round__mono,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y3 )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).

% round_mono
thf(fact_7877_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).

% summable_norm
thf(fact_7878_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).

% summable_norm
thf(fact_7879_ceiling__ge__round,axiom,
    ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).

% ceiling_ge_round
thf(fact_7880_tanh__real__gt__neg1,axiom,
    ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).

% tanh_real_gt_neg1
thf(fact_7881_powser__inside,axiom,
    ! [F: nat > real,X2: real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).

% powser_inside
thf(fact_7882_powser__inside,axiom,
    ! [F: nat > complex,X2: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
       => ( summable_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).

% powser_inside
thf(fact_7883_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
     => ( ( suminf_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7884_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7885_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7886_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7887_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N5: nat] :
            ( ( ord_less_eq_nat @ N9 @ N5 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7888_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N5: nat] :
            ( ( ord_less_eq_nat @ N9 @ N5 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7889_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_7890_Abel__lemma,axiom,
    ! [R2: real,R0: real,A2: nat > complex,M5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M5 )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_7891_summable__ratio__test,axiom,
    ! [C: real,N6: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_7892_summable__ratio__test,axiom,
    ! [C: real,N6: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_7893_round__diff__minimal,axiom,
    ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_7894_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_7895_of__int__round__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7896_of__int__round__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7897_of__int__round__ge,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).

% of_int_round_ge
thf(fact_7898_of__int__round__ge,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).

% of_int_round_ge
thf(fact_7899_of__int__round__gt,axiom,
    ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).

% of_int_round_gt
thf(fact_7900_of__int__round__gt,axiom,
    ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).

% of_int_round_gt
thf(fact_7901_of__int__round__abs__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7902_of__int__round__abs__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7903_accp__subset,axiom,
    ! [R1: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,R22: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o] :
      ( ( ord_le7862513914298786254_nat_o @ R1 @ R22 )
     => ( ord_le8126618931240741628_nat_o @ ( accp_P8646395344606611882on_nat @ R22 ) @ ( accp_P8646395344606611882on_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7904_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7905_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_7906_accp__subset,axiom,
    ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
      ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
     => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7907_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7908_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X2: $o > complex,Y3: $o > complex] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7909_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > complex,Y3: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7910_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7911_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > complex,Y3: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7912_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7913_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X2: extended_enat > complex,Y3: extended_enat > complex] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7914_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X2: $o > real,Y3: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7915_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > real,Y3: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7916_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7917_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > real,Y3: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7918_sum__gp,axiom,
    ! [N: nat,M: nat,X2: rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X2 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7919_sum__gp,axiom,
    ! [N: nat,M: nat,X2: complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X2 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7920_sum__gp,axiom,
    ! [N: nat,M: nat,X2: real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X2 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7921_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X2: $o > real,Y3: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7922_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > real,Y3: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7923_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7924_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > real,Y3: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7925_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7926_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Extended_enat,X2: extended_enat > real,Y3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I4: extended_enat] :
              ( ( member_Extended_enat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I4: extended_enat] :
                ( ( member_Extended_enat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7927_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X2: $o > rat,Y3: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7928_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > rat,Y3: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7929_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7930_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > rat,Y3: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7931_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_7932_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_7933_log__base__10__eq1,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
        = ( 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 @ X2 ) ) ) ) ).

% log_base_10_eq1
thf(fact_7934_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_7935_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_7936_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A3 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_7937_sum_Oneutral__const,axiom,
    ! [A3: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A3 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_7938_sum_Oneutral__const,axiom,
    ! [A3: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A3 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_7939_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A3 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_7940_sums__zero,axiom,
    ( sums_real
    @ ^ [N2: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_7941_sums__zero,axiom,
    ( sums_nat
    @ ^ [N2: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_7942_sums__zero,axiom,
    ( sums_int
    @ ^ [N2: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_7943_sum_Oempty,axiom,
    ! [G3: real > real] :
      ( ( groups8097168146408367636l_real @ G3 @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7944_sum_Oempty,axiom,
    ! [G3: real > rat] :
      ( ( groups1300246762558778688al_rat @ G3 @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7945_sum_Oempty,axiom,
    ! [G3: real > nat] :
      ( ( groups1935376822645274424al_nat @ G3 @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7946_sum_Oempty,axiom,
    ! [G3: real > int] :
      ( ( groups1932886352136224148al_int @ G3 @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7947_sum_Oempty,axiom,
    ! [G3: $o > real] :
      ( ( groups8691415230153176458o_real @ G3 @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7948_sum_Oempty,axiom,
    ! [G3: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G3 @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7949_sum_Oempty,axiom,
    ! [G3: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G3 @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7950_sum_Oempty,axiom,
    ! [G3: $o > int] :
      ( ( groups8505340233167759370_o_int @ G3 @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7951_sum_Oempty,axiom,
    ! [G3: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G3 @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7952_sum_Oempty,axiom,
    ! [G3: nat > int] :
      ( ( groups3539618377306564664at_int @ G3 @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7953_sum_Oinfinite,axiom,
    ! [A3: set_int,G3: int > real] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G3 @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7954_sum_Oinfinite,axiom,
    ! [A3: set_complex,G3: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G3 @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7955_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G3 @ A3 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7956_sum_Oinfinite,axiom,
    ! [A3: set_nat,G3: nat > rat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G3 @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7957_sum_Oinfinite,axiom,
    ! [A3: set_int,G3: int > rat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7958_sum_Oinfinite,axiom,
    ! [A3: set_complex,G3: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7959_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > rat] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7960_sum_Oinfinite,axiom,
    ! [A3: set_int,G3: int > nat] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7961_sum_Oinfinite,axiom,
    ! [A3: set_complex,G3: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7962_sum_Oinfinite,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > nat] :
      ( ~ ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7963_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: int > nat] :
      ( ( finite_finite_int @ F2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X: int] :
              ( ( member_int @ X @ F2 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7964_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ F2 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7965_sum__eq__0__iff,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ F2 )
     => ( ( ( groups977919841031483927at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ F2 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7966_sum__eq__0__iff,axiom,
    ! [F2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ F2 )
     => ( ( ( groups2027974829824023292at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X: extended_enat] :
              ( ( member_Extended_enat @ X @ F2 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7967_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ F2 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7968_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_7969_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_7970_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_7971_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_7972_sum_Odelta_H,axiom,
    ! [S: set_real,A2: real,B3: real > real] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A2 @ S )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_real @ A2 @ S )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7973_sum_Odelta_H,axiom,
    ! [S: set_o,A2: $o,B3: $o > real] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A2 @ S )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_o @ A2 @ S )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7974_sum_Odelta_H,axiom,
    ! [S: set_int,A2: int,B3: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A2 @ S )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_int @ A2 @ S )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7975_sum_Odelta_H,axiom,
    ! [S: set_complex,A2: complex,B3: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7976_sum_Odelta_H,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ( member_Extended_enat @ A2 @ S )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ S )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7977_sum_Odelta_H,axiom,
    ! [S: set_real,A2: real,B3: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A2 @ S )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_real @ A2 @ S )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7978_sum_Odelta_H,axiom,
    ! [S: set_o,A2: $o,B3: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A2 @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_o @ A2 @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7979_sum_Odelta_H,axiom,
    ! [S: set_nat,A2: nat,B3: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A2 @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_nat @ A2 @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7980_sum_Odelta_H,axiom,
    ! [S: set_int,A2: int,B3: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A2 @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_int @ A2 @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7981_sum_Odelta_H,axiom,
    ! [S: set_complex,A2: complex,B3: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7982_sum_Odelta,axiom,
    ! [S: set_real,A2: real,B3: real > real] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A2 @ S )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_real @ A2 @ S )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7983_sum_Odelta,axiom,
    ! [S: set_o,A2: $o,B3: $o > real] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A2 @ S )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_o @ A2 @ S )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7984_sum_Odelta,axiom,
    ! [S: set_int,A2: int,B3: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A2 @ S )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_int @ A2 @ S )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7985_sum_Odelta,axiom,
    ! [S: set_complex,A2: complex,B3: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7986_sum_Odelta,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ( member_Extended_enat @ A2 @ S )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ S )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7987_sum_Odelta,axiom,
    ! [S: set_real,A2: real,B3: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A2 @ S )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_real @ A2 @ S )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7988_sum_Odelta,axiom,
    ! [S: set_o,A2: $o,B3: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( ( member_o @ A2 @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_o @ A2 @ S )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7989_sum_Odelta,axiom,
    ! [S: set_nat,A2: nat,B3: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A2 @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_nat @ A2 @ S )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7990_sum_Odelta,axiom,
    ! [S: set_int,A2: int,B3: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( ( member_int @ A2 @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_int @ A2 @ S )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7991_sum_Odelta,axiom,
    ! [S: set_complex,A2: complex,B3: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = ( B3 @ A2 ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7992_sum__abs,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_7993_sum__abs,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A3 ) ) ).

% sum_abs
thf(fact_7994_sum_Oinsert,axiom,
    ! [A3: set_real,X2: real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ~ ( member_real @ X2 @ A3 )
       => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_7995_sum_Oinsert,axiom,
    ! [A3: set_o,X2: $o,G3: $o > real] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X2 @ A3 )
       => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_7996_sum_Oinsert,axiom,
    ! [A3: set_int,X2: int,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ~ ( member_int @ X2 @ A3 )
       => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_7997_sum_Oinsert,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ~ ( member_complex @ X2 @ A3 )
       => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_7998_sum_Oinsert,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ~ ( member_Extended_enat @ X2 @ A3 )
       => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_7999_sum_Oinsert,axiom,
    ! [A3: set_real,X2: real,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ~ ( member_real @ X2 @ A3 )
       => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_8000_sum_Oinsert,axiom,
    ! [A3: set_o,X2: $o,G3: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X2 @ A3 )
       => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_8001_sum_Oinsert,axiom,
    ! [A3: set_nat,X2: nat,G3: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat @ X2 @ A3 )
       => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_8002_sum_Oinsert,axiom,
    ! [A3: set_int,X2: int,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ~ ( member_int @ X2 @ A3 )
       => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_8003_sum_Oinsert,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ~ ( member_complex @ X2 @ A3 )
       => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).

% sum.insert
thf(fact_8004_sum__abs__ge__zero,axiom,
    ! [F: int > int,A3: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_8005_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A3 ) ) ).

% sum_abs_ge_zero
thf(fact_8006_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_8007_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_8008_powser__sums__zero__iff,axiom,
    ! [A2: nat > complex,X2: complex] :
      ( ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( A2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
        @ X2 )
      = ( ( A2 @ zero_zero_nat )
        = X2 ) ) ).

% powser_sums_zero_iff
thf(fact_8009_powser__sums__zero__iff,axiom,
    ! [A2: nat > real,X2: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( A2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
        @ X2 )
      = ( ( A2 @ zero_zero_nat )
        = X2 ) ) ).

% powser_sums_zero_iff
thf(fact_8010_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8011_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8012_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8013_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G3: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8014_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A3 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_8015_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8016_sum__zero__power,axiom,
    ! [A3: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A3 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A3 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_8017_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A3 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A3 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_8018_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A3 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A3 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_8019_sum__zero__power_H,axiom,
    ! [A3: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A3 )
          & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A3 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A3 )
            & ( member_nat @ zero_zero_nat @ A3 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A3 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_8020_sums__le,axiom,
    ! [F: nat > nat,G3: nat > nat,S2: nat,T: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G3 @ N3 ) )
     => ( ( sums_nat @ F @ S2 )
       => ( ( sums_nat @ G3 @ T )
         => ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_8021_sums__le,axiom,
    ! [F: nat > int,G3: nat > int,S2: int,T: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G3 @ N3 ) )
     => ( ( sums_int @ F @ S2 )
       => ( ( sums_int @ G3 @ T )
         => ( ord_less_eq_int @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_8022_sums__0,axiom,
    ! [F: nat > real] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_real )
     => ( sums_real @ F @ zero_zero_real ) ) ).

% sums_0
thf(fact_8023_sums__0,axiom,
    ! [F: nat > nat] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_nat )
     => ( sums_nat @ F @ zero_zero_nat ) ) ).

% sums_0
thf(fact_8024_sums__0,axiom,
    ! [F: nat > int] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_int )
     => ( sums_int @ F @ zero_zero_int ) ) ).

% sums_0
thf(fact_8025_sum_Oneutral,axiom,
    ! [A3: set_nat,G3: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ( ( G3 @ X5 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G3 @ A3 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_8026_sum_Oneutral,axiom,
    ! [A3: set_complex,G3: complex > complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A3 )
         => ( ( G3 @ X5 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G3 @ A3 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_8027_sum_Oneutral,axiom,
    ! [A3: set_int,G3: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ( G3 @ X5 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G3 @ A3 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_8028_sum_Oneutral,axiom,
    ! [A3: set_nat,G3: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ( ( G3 @ X5 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G3 @ A3 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_8029_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: real > real,A3: set_real] :
      ( ( ( groups8097168146408367636l_real @ G3 @ A3 )
       != zero_zero_real )
     => ~ ! [A: real] :
            ( ( member_real @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8030_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: $o > real,A3: set_o] :
      ( ( ( groups8691415230153176458o_real @ G3 @ A3 )
       != zero_zero_real )
     => ~ ! [A: $o] :
            ( ( member_o @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8031_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: int > real,A3: set_int] :
      ( ( ( groups8778361861064173332t_real @ G3 @ A3 )
       != zero_zero_real )
     => ~ ! [A: int] :
            ( ( member_int @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8032_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: real > rat,A3: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G3 @ A3 )
       != zero_zero_rat )
     => ~ ! [A: real] :
            ( ( member_real @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8033_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: $o > rat,A3: set_o] :
      ( ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
       != zero_zero_rat )
     => ~ ! [A: $o] :
            ( ( member_o @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8034_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: nat > rat,A3: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G3 @ A3 )
       != zero_zero_rat )
     => ~ ! [A: nat] :
            ( ( member_nat @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8035_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: int > rat,A3: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
       != zero_zero_rat )
     => ~ ! [A: int] :
            ( ( member_int @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8036_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: real > nat,A3: set_real] :
      ( ( ( groups1935376822645274424al_nat @ G3 @ A3 )
       != zero_zero_nat )
     => ~ ! [A: real] :
            ( ( member_real @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8037_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: $o > nat,A3: set_o] :
      ( ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
       != zero_zero_nat )
     => ~ ! [A: $o] :
            ( ( member_o @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8038_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G3: int > nat,A3: set_int] :
      ( ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
       != zero_zero_nat )
     => ~ ! [A: int] :
            ( ( member_int @ A @ A3 )
           => ( ( G3 @ A )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8039_sum__norm__le,axiom,
    ! [S: set_real,F: real > complex,G3: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8040_sum__norm__le,axiom,
    ! [S: set_o,F: $o > complex,G3: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S ) ) @ ( groups8691415230153176458o_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8041_sum__norm__le,axiom,
    ! [S: set_set_nat,F: set_nat > complex,G3: set_nat > real] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( groups5107569545109728110t_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8042_sum__norm__le,axiom,
    ! [S: set_int,F: int > complex,G3: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( groups8778361861064173332t_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8043_sum__norm__le,axiom,
    ! [S: set_nat,F: nat > complex,G3: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8044_sum__norm__le,axiom,
    ! [S: set_complex,F: complex > complex,G3: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8045_sum__norm__le,axiom,
    ! [S: set_nat,F: nat > real,G3: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G3 @ S ) ) ) ).

% sum_norm_le
thf(fact_8046_sum__mono,axiom,
    ! [K4: set_real,F: real > rat,G3: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8047_sum__mono,axiom,
    ! [K4: set_o,F: $o > rat,G3: $o > rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ K4 ) @ ( groups7872700643590313910_o_rat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8048_sum__mono,axiom,
    ! [K4: set_nat,F: nat > rat,G3: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8049_sum__mono,axiom,
    ! [K4: set_int,F: int > rat,G3: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8050_sum__mono,axiom,
    ! [K4: set_real,F: real > nat,G3: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8051_sum__mono,axiom,
    ! [K4: set_o,F: $o > nat,G3: $o > nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K4 ) @ ( groups8507830703676809646_o_nat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8052_sum__mono,axiom,
    ! [K4: set_int,F: int > nat,G3: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K4 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8053_sum__mono,axiom,
    ! [K4: set_real,F: real > int,G3: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8054_sum__mono,axiom,
    ! [K4: set_o,F: $o > int,G3: $o > int] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K4 ) @ ( groups8505340233167759370_o_int @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8055_sum__mono,axiom,
    ! [K4: set_nat,F: nat > int,G3: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K4 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G3 @ K4 ) ) ) ).

% sum_mono
thf(fact_8056_sum_Oswap__restrict,axiom,
    ! [A3: set_o,B2: set_nat,G3: $o > nat > nat,R: $o > nat > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups8507830703676809646_o_nat
            @ ^ [X: $o] :
                ( groups3542108847815614940at_nat @ ( G3 @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups8507830703676809646_o_nat
                @ ^ [X: $o] : ( G3 @ X @ Y )
                @ ( collect_o
                  @ ^ [X: $o] :
                      ( ( member_o @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8057_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B2: set_nat,G3: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X: real] :
                ( groups3542108847815614940at_nat @ ( G3 @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X: real] : ( G3 @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8058_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B2: set_nat,G3: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X: int] :
                ( groups3542108847815614940at_nat @ ( G3 @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X: int] : ( G3 @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8059_sum_Oswap__restrict,axiom,
    ! [A3: set_complex,B2: set_nat,G3: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X: complex] :
                ( groups3542108847815614940at_nat @ ( G3 @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X: complex] : ( G3 @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8060_sum_Oswap__restrict,axiom,
    ! [A3: set_Extended_enat,B2: set_nat,G3: extended_enat > nat > nat,R: extended_enat > nat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2027974829824023292at_nat
            @ ^ [X: extended_enat] :
                ( groups3542108847815614940at_nat @ ( G3 @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups2027974829824023292at_nat
                @ ^ [X: extended_enat] : ( G3 @ X @ Y )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X: extended_enat] :
                      ( ( member_Extended_enat @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8061_sum_Oswap__restrict,axiom,
    ! [A3: set_o,B2: set_complex,G3: $o > complex > complex,R: $o > complex > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5328290441151304332omplex
            @ ^ [X: $o] :
                ( groups7754918857620584856omplex @ ( G3 @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups5328290441151304332omplex
                @ ^ [X: $o] : ( G3 @ X @ Y )
                @ ( collect_o
                  @ ^ [X: $o] :
                      ( ( member_o @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8062_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B2: set_complex,G3: real > complex > complex,R: real > complex > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X: real] :
                ( groups7754918857620584856omplex @ ( G3 @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X: real] : ( G3 @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8063_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B2: set_complex,G3: nat > complex > complex,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X: nat] :
                ( groups7754918857620584856omplex @ ( G3 @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X: nat] : ( G3 @ X @ Y )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8064_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B2: set_complex,G3: int > complex > complex,R: int > complex > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X: int] :
                ( groups7754918857620584856omplex @ ( G3 @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X: int] : ( G3 @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8065_sum_Oswap__restrict,axiom,
    ! [A3: set_Extended_enat,B2: set_complex,G3: extended_enat > complex > complex,R: extended_enat > complex > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups6818542070133387226omplex
            @ ^ [X: extended_enat] :
                ( groups7754918857620584856omplex @ ( G3 @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A3 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups6818542070133387226omplex
                @ ^ [X: extended_enat] : ( G3 @ X @ Y )
                @ ( collec4429806609662206161d_enat
                  @ ^ [X: extended_enat] :
                      ( ( member_Extended_enat @ X @ A3 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8066_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).

% sums_If_finite_set
thf(fact_8067_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).

% sums_If_finite_set
thf(fact_8068_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A3 )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).

% sums_If_finite_set
thf(fact_8069_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8070_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8071_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8072_sums__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_8073_sums__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_8074_sums__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).

% sums_finite
thf(fact_8075_norm__sum,axiom,
    ! [F: nat > complex,A3: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A3 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A3 ) ) ).

% norm_sum
thf(fact_8076_norm__sum,axiom,
    ! [F: complex > complex,A3: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A3 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A3 ) ) ).

% norm_sum
thf(fact_8077_norm__sum,axiom,
    ! [F: nat > real,A3: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
        @ A3 ) ) ).

% norm_sum
thf(fact_8078_sum__nonpos,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8079_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8080_sum__nonpos,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8081_sum__nonpos,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8082_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8083_sum__nonpos,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8084_sum__nonpos,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8085_sum__nonpos,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8086_sum__nonpos,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8087_sum__nonpos,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8088_sum__nonneg,axiom,
    ! [A3: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8089_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8090_sum__nonneg,axiom,
    ! [A3: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8091_sum__nonneg,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8092_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8093_sum__nonneg,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8094_sum__nonneg,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8095_sum__nonneg,axiom,
    ! [A3: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8096_sum__nonneg,axiom,
    ! [A3: set_o,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8097_sum__nonneg,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).

% sum_nonneg
thf(fact_8098_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G3: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G3 @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8099_sum__mono__inv,axiom,
    ! [F: $o > rat,I5: set_o,G3: $o > rat,I: $o] :
      ( ( ( groups7872700643590313910_o_rat @ F @ I5 )
        = ( groups7872700643590313910_o_rat @ G3 @ I5 ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8100_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G3: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G3 @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8101_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G3: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G3 @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8102_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G3: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G3 @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8103_sum__mono__inv,axiom,
    ! [F: extended_enat > rat,I5: set_Extended_enat,G3: extended_enat > rat,I: extended_enat] :
      ( ( ( groups1392844769737527556at_rat @ F @ I5 )
        = ( groups1392844769737527556at_rat @ G3 @ I5 ) )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_Extended_enat @ I @ I5 )
         => ( ( finite4001608067531595151d_enat @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8104_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G3: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G3 @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8105_sum__mono__inv,axiom,
    ! [F: $o > nat,I5: set_o,G3: $o > nat,I: $o] :
      ( ( ( groups8507830703676809646_o_nat @ F @ I5 )
        = ( groups8507830703676809646_o_nat @ G3 @ I5 ) )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8106_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G3: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G3 @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8107_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G3: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G3 @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G3 @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8108_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: nat > nat,G3: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A3 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G3 @ ( suc @ X5 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A3 )
          = ( groups3542108847815614940at_nat @ G3 @ A3 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8109_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: nat > real,G3: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A3 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G3 @ ( suc @ X5 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A3 )
          = ( groups6591440286371151544t_real @ G3 @ A3 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8110_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8111_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8112_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8113_sum_Ointer__filter,axiom,
    ! [A3: set_o,G3: $o > real,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups8691415230153176458o_real @ G3
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups8691415230153176458o_real
          @ ^ [X: $o] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8114_sum_Ointer__filter,axiom,
    ! [A3: set_real,G3: real > real,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G3
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8115_sum_Ointer__filter,axiom,
    ! [A3: set_int,G3: int > real,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G3
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8116_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G3: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G3
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8117_sum_Ointer__filter,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > real,P: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G3
          @ ( collec4429806609662206161d_enat
            @ ^ [X: extended_enat] :
                ( ( member_Extended_enat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups4148127829035722712t_real
          @ ^ [X: extended_enat] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8118_sum_Ointer__filter,axiom,
    ! [A3: set_o,G3: $o > rat,P: $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups7872700643590313910_o_rat @ G3
          @ ( collect_o
            @ ^ [X: $o] :
                ( ( member_o @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X: $o] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8119_sum_Ointer__filter,axiom,
    ! [A3: set_real,G3: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat @ G3
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8120_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G3: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups2906978787729119204at_rat @ G3
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8121_sum_Ointer__filter,axiom,
    ! [A3: set_int,G3: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G3
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8122_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G3: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G3
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_8123_sums__If__finite__set_H,axiom,
    ! [G3: nat > real,S: real,A3: set_nat,S5: real,F: nat > real] :
      ( ( sums_real @ G3 @ S )
     => ( ( finite_finite_nat @ A3 )
       => ( ( S5
            = ( plus_plus_real @ S
              @ ( groups6591440286371151544t_real
                @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G3 @ N2 ) )
                @ A3 ) ) )
         => ( sums_real
            @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A3 ) @ ( F @ N2 ) @ ( G3 @ N2 ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_8124_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G3: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8125_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G3: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8126_sum__le__included,axiom,
    ! [S2: set_int,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > int,F: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8127_sum__le__included,axiom,
    ! [S2: set_complex,T: set_int,G3: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8128_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G3: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8129_sum__le__included,axiom,
    ! [S2: set_complex,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8130_sum__le__included,axiom,
    ! [S2: set_Extended_enat,T: set_int,G3: int > real,I: int > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8131_sum__le__included,axiom,
    ! [S2: set_Extended_enat,T: set_complex,G3: complex > real,I: complex > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8132_sum__le__included,axiom,
    ! [S2: set_Extended_enat,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S2 )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8133_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G3: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G3 @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G3 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8134_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X: real] :
                ( ( member_real @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8135_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: $o > real] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X: $o] :
                ( ( member_o @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8136_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X: int] :
                ( ( member_int @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8137_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8138_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ A3 )
            = zero_zero_real )
          = ( ! [X: extended_enat] :
                ( ( member_Extended_enat @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8139_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8140_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X: $o] :
                ( ( member_o @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8141_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8142_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8143_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A3 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A3 )
            = zero_zero_rat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A3 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8144_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > real,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A3 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8145_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: complex > real,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A3 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A3 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8146_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A3 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8147_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: nat > rat,G3: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8148_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > rat,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A3 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8149_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: complex > rat,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A3 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8150_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A3 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8151_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: int > nat,G3: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A3 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8152_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: complex > nat,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A3 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8153_sum__strict__mono__ex1,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A3 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
       => ( ? [X4: extended_enat] :
              ( ( member_Extended_enat @ X4 @ A3 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8154_sum_Orelated,axiom,
    ! [R: real > real > $o,S: set_int,H2: int > real,G3: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S ) @ ( groups8778361861064173332t_real @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8155_sum_Orelated,axiom,
    ! [R: real > real > $o,S: set_complex,H2: complex > real,G3: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S ) @ ( groups5808333547571424918x_real @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8156_sum_Orelated,axiom,
    ! [R: real > real > $o,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups4148127829035722712t_real @ H2 @ S ) @ ( groups4148127829035722712t_real @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8157_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_nat,H2: nat > rat,G3: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H2 @ S ) @ ( groups2906978787729119204at_rat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8158_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_int,H2: int > rat,G3: int > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S ) @ ( groups3906332499630173760nt_rat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8159_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_complex,H2: complex > rat,G3: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S ) @ ( groups5058264527183730370ex_rat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8160_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups1392844769737527556at_rat @ H2 @ S ) @ ( groups1392844769737527556at_rat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8161_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S: set_int,H2: int > nat,G3: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
       => ( ( finite_finite_int @ S )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S ) @ ( groups4541462559716669496nt_nat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8162_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S: set_complex,H2: complex > nat,G3: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S ) @ ( groups5693394587270226106ex_nat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8163_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S: set_Extended_enat,H2: extended_enat > nat,G3: extended_enat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y22 ) )
           => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
       => ( ( finite4001608067531595151d_enat @ S )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
           => ( R @ ( groups2027974829824023292at_nat @ H2 @ S ) @ ( groups2027974829824023292at_nat @ G3 @ S ) ) ) ) ) ) ).

% sum.related
thf(fact_8164_sum__strict__mono,axiom,
    ! [A3: set_complex,F: complex > real,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8165_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ A3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8166_sum__strict__mono,axiom,
    ! [A3: set_real,F: real > real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8167_sum__strict__mono,axiom,
    ! [A3: set_o,F: $o > real,G3: $o > real] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8168_sum__strict__mono,axiom,
    ! [A3: set_int,F: int > real,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8169_sum__strict__mono,axiom,
    ! [A3: set_complex,F: complex > rat,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8170_sum__strict__mono,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ A3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8171_sum__strict__mono,axiom,
    ! [A3: set_real,F: real > rat,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8172_sum__strict__mono,axiom,
    ! [A3: set_o,F: $o > rat,G3: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8173_sum__strict__mono,axiom,
    ! [A3: set_nat,F: nat > rat,G3: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8174_sum_Oinsert__if,axiom,
    ! [A3: set_real,X2: real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( member_real @ X2 @ A3 )
         => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
            = ( groups8097168146408367636l_real @ G3 @ A3 ) ) )
        & ( ~ ( member_real @ X2 @ A3 )
         => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
            = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8175_sum_Oinsert__if,axiom,
    ! [A3: set_o,X2: $o,G3: $o > real] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( member_o @ X2 @ A3 )
         => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
            = ( groups8691415230153176458o_real @ G3 @ A3 ) ) )
        & ( ~ ( member_o @ X2 @ A3 )
         => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
            = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8176_sum_Oinsert__if,axiom,
    ! [A3: set_int,X2: int,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( member_int @ X2 @ A3 )
         => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
            = ( groups8778361861064173332t_real @ G3 @ A3 ) ) )
        & ( ~ ( member_int @ X2 @ A3 )
         => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
            = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8177_sum_Oinsert__if,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( member_complex @ X2 @ A3 )
         => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
            = ( groups5808333547571424918x_real @ G3 @ A3 ) ) )
        & ( ~ ( member_complex @ X2 @ A3 )
         => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
            = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8178_sum_Oinsert__if,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( member_Extended_enat @ X2 @ A3 )
         => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
            = ( groups4148127829035722712t_real @ G3 @ A3 ) ) )
        & ( ~ ( member_Extended_enat @ X2 @ A3 )
         => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
            = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8179_sum_Oinsert__if,axiom,
    ! [A3: set_real,X2: real,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( member_real @ X2 @ A3 )
         => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
            = ( groups1300246762558778688al_rat @ G3 @ A3 ) ) )
        & ( ~ ( member_real @ X2 @ A3 )
         => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
            = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8180_sum_Oinsert__if,axiom,
    ! [A3: set_o,X2: $o,G3: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( member_o @ X2 @ A3 )
         => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
            = ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) )
        & ( ~ ( member_o @ X2 @ A3 )
         => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
            = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8181_sum_Oinsert__if,axiom,
    ! [A3: set_nat,X2: nat,G3: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( member_nat @ X2 @ A3 )
         => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
            = ( groups2906978787729119204at_rat @ G3 @ A3 ) ) )
        & ( ~ ( member_nat @ X2 @ A3 )
         => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
            = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8182_sum_Oinsert__if,axiom,
    ! [A3: set_int,X2: int,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( member_int @ X2 @ A3 )
         => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
            = ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) )
        & ( ~ ( member_int @ X2 @ A3 )
         => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
            = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8183_sum_Oinsert__if,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( member_complex @ X2 @ A3 )
         => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
            = ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) )
        & ( ~ ( member_complex @ X2 @ A3 )
         => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
            = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8184_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S: set_real,I: real > real,J: real > real,T2: set_real,G3: real > real,H2: real > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
               => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T2 @ T5 ) ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: real] :
                    ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( ( member_real @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G3 @ S )
                        = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8185_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_o,S: set_real,I: $o > real,J: real > $o,T2: set_o,G3: real > real,H2: $o > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
               => ( member_o @ ( J @ A ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: $o] :
                    ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: $o] :
                        ( ( member_o @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G3 @ S )
                        = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8186_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_real,S: set_o,I: real > $o,J: $o > real,T2: set_real,G3: $o > real,H2: real > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A: $o] :
              ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
               => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T2 @ T5 ) ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: real] :
                    ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
                   => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
               => ( ! [A: $o] :
                      ( ( member_o @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( ( member_real @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: $o] :
                          ( ( member_o @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8691415230153176458o_real @ G3 @ S )
                        = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8187_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_o,S: set_o,I: $o > $o,J: $o > $o,T2: set_o,G3: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_o @ T5 )
       => ( ! [A: $o] :
              ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
               => ( member_o @ ( J @ A ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: $o] :
                    ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
                   => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
               => ( ! [A: $o] :
                      ( ( member_o @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: $o] :
                        ( ( member_o @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: $o] :
                          ( ( member_o @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8691415230153176458o_real @ G3 @ S )
                        = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8188_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S: set_real,I: int > real,J: real > int,T2: set_int,G3: real > real,H2: int > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
               => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: int] :
                    ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( ( member_int @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G3 @ S )
                        = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8189_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_int,S: set_o,I: int > $o,J: $o > int,T2: set_int,G3: $o > real,H2: int > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A: $o] :
              ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
               => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: int] :
                    ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
                   => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
               => ( ! [A: $o] :
                      ( ( member_o @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( ( member_int @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: $o] :
                          ( ( member_o @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8691415230153176458o_real @ G3 @ S )
                        = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8190_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S: set_real,I: complex > real,J: real > complex,T2: set_complex,G3: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
               => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T2 @ T5 ) ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: complex] :
                    ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( ( member_complex @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G3 @ S )
                        = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8191_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_complex,S: set_o,I: complex > $o,J: $o > complex,T2: set_complex,G3: $o > real,H2: complex > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A: $o] :
              ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
               => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T2 @ T5 ) ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: complex] :
                    ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
                   => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
               => ( ! [A: $o] :
                      ( ( member_o @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( ( member_complex @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: $o] :
                          ( ( member_o @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8691415230153176458o_real @ G3 @ S )
                        = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8192_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_Extended_enat,S: set_real,I: extended_enat > real,J: real > extended_enat,T2: set_Extended_enat,G3: real > real,H2: extended_enat > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
               => ( member_Extended_enat @ ( J @ A ) @ ( minus_925952699566721837d_enat @ T2 @ T5 ) ) )
           => ( ! [B: extended_enat] :
                  ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: extended_enat] :
                    ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: extended_enat] :
                        ( ( member_Extended_enat @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G3 @ S )
                        = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8193_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T5: set_Extended_enat,S: set_o,I: extended_enat > $o,J: $o > extended_enat,T2: set_Extended_enat,G3: $o > real,H2: extended_enat > real] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite4001608067531595151d_enat @ T5 )
       => ( ! [A: $o] :
              ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
               => ( member_Extended_enat @ ( J @ A ) @ ( minus_925952699566721837d_enat @ T2 @ T5 ) ) )
           => ( ! [B: extended_enat] :
                  ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: extended_enat] :
                    ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
                   => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
               => ( ! [A: $o] :
                      ( ( member_o @ A @ S5 )
                     => ( ( G3 @ A )
                        = zero_zero_real ) )
                 => ( ! [B: extended_enat] :
                        ( ( member_Extended_enat @ B @ T5 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: $o] :
                          ( ( member_o @ A @ S )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G3 @ A ) ) )
                     => ( ( groups8691415230153176458o_real @ G3 @ S )
                        = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8194_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8195_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8196_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8197_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8198_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: real > real,B2: real,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S2 )
            = B2 )
         => ( ( member_real @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8199_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > real,B2: real,I: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S2 )
            = B2 )
         => ( ( member_o @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8200_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > real,B2: real,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S2 )
            = B2 )
         => ( ( member_int @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8201_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: complex > real,B2: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S2 )
            = B2 )
         => ( ( member_complex @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8202_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Extended_enat,F: extended_enat > real,B2: real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S2 )
            = B2 )
         => ( ( member_Extended_enat @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8203_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: real > rat,B2: rat,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
            = B2 )
         => ( ( member_real @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8204_sum__nonneg__leq__bound,axiom,
    ! [S2: set_o,F: $o > rat,B2: rat,I: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
            = B2 )
         => ( ( member_o @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8205_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: nat > rat,B2: rat,I: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
            = B2 )
         => ( ( member_nat @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8206_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > rat,B2: rat,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
            = B2 )
         => ( ( member_int @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8207_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: complex > rat,B2: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
            = B2 )
         => ( ( member_complex @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8208_sum__nonneg__0,axiom,
    ! [S2: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_real @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8209_sum__nonneg__0,axiom,
    ! [S2: set_o,F: $o > real,I: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_o @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8210_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_int @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8211_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_complex @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8212_sum__nonneg__0,axiom,
    ! [S2: set_Extended_enat,F: extended_enat > real,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S2 )
     => ( ! [I2: extended_enat] :
            ( ( member_Extended_enat @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups4148127829035722712t_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_Extended_enat @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8213_sum__nonneg__0,axiom,
    ! [S2: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_real @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8214_sum__nonneg__0,axiom,
    ! [S2: set_o,F: $o > rat,I: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_o @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8215_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8216_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_int @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8217_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8218_sum_Ointer__restrict,axiom,
    ! [A3: set_real,G3: real > real,B2: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G3 @ ( inf_inf_set_real @ A3 @ B2 ) )
        = ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( member_real @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8219_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G3: $o > real,B2: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups8691415230153176458o_real @ G3 @ ( inf_inf_set_o @ A3 @ B2 ) )
        = ( groups8691415230153176458o_real
          @ ^ [X: $o] : ( if_real @ ( member_o @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8220_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G3: int > real,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) )
        = ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( member_int @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8221_sum_Ointer__restrict,axiom,
    ! [A3: set_complex,G3: complex > real,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) )
        = ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( member_complex @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8222_sum_Ointer__restrict,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > real,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
        = ( groups4148127829035722712t_real
          @ ^ [X: extended_enat] : ( if_real @ ( member_Extended_enat @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8223_sum_Ointer__restrict,axiom,
    ! [A3: set_real,G3: real > rat,B2: set_real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat @ G3 @ ( inf_inf_set_real @ A3 @ B2 ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X: real] : ( if_rat @ ( member_real @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8224_sum_Ointer__restrict,axiom,
    ! [A3: set_o,G3: $o > rat,B2: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( groups7872700643590313910_o_rat @ G3 @ ( inf_inf_set_o @ A3 @ B2 ) )
        = ( groups7872700643590313910_o_rat
          @ ^ [X: $o] : ( if_rat @ ( member_o @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8225_sum_Ointer__restrict,axiom,
    ! [A3: set_int,G3: int > rat,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X: int] : ( if_rat @ ( member_int @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8226_sum_Ointer__restrict,axiom,
    ! [A3: set_complex,G3: complex > rat,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X: complex] : ( if_rat @ ( member_complex @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8227_sum_Ointer__restrict,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > rat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
        = ( groups1392844769737527556at_rat
          @ ^ [X: extended_enat] : ( if_rat @ ( member_Extended_enat @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_8228_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G3
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G3 @ X )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8229_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G3
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G3 @ X )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8230_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G3
          @ ( minus_811609699411566653omplex @ A3
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G3 @ X )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8231_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G3
          @ ( minus_925952699566721837d_enat @ A3
            @ ( collec4429806609662206161d_enat
              @ ^ [X: extended_enat] :
                  ( ( G3 @ X )
                  = zero_zero_real ) ) ) )
        = ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8232_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat @ G3
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G3 @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8233_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G3
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G3 @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8234_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G3
          @ ( minus_811609699411566653omplex @ A3
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G3 @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8235_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G3
          @ ( minus_925952699566721837d_enat @ A3
            @ ( collec4429806609662206161d_enat
              @ ^ [X: extended_enat] :
                  ( ( G3 @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8236_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G3: real > nat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1935376822645274424al_nat @ G3
          @ ( minus_minus_set_real @ A3
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G3 @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8237_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G3: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G3
          @ ( minus_minus_set_int @ A3
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G3 @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8238_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G3: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8239_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G3: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8240_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G3: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G3 @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8241_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G3: nat > real,M: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G3 @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8242_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8243_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8244_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8245_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8246_sum__pos2,axiom,
    ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( member_Extended_enat @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8247_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8248_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8249_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8250_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8251_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8252_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8253_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8254_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8255_sum__pos,axiom,
    ! [I5: set_o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8256_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8257_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8258_sum__pos,axiom,
    ! [I5: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ I5 )
     => ( ( I5 != bot_bo7653980558646680370d_enat )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8259_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8260_sum__pos,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8261_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8262_sum__bounded__below,axiom,
    ! [A3: set_real,K4: real,F: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8263_sum__bounded__below,axiom,
    ! [A3: set_o,K4: real,F: $o > real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) @ K4 ) @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8264_sum__bounded__below,axiom,
    ! [A3: set_complex,K4: real,F: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) @ K4 ) @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8265_sum__bounded__below,axiom,
    ! [A3: set_int,K4: real,F: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) @ K4 ) @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8266_sum__bounded__below,axiom,
    ! [A3: set_real,K4: rat,F: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8267_sum__bounded__below,axiom,
    ! [A3: set_o,K4: rat,F: $o > rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8268_sum__bounded__below,axiom,
    ! [A3: set_complex,K4: rat,F: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8269_sum__bounded__below,axiom,
    ! [A3: set_nat,K4: rat,F: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8270_sum__bounded__below,axiom,
    ! [A3: set_int,K4: rat,F: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8271_sum__bounded__below,axiom,
    ! [A3: set_real,K4: nat,F: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_nat @ K4 @ ( F @ I2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).

% sum_bounded_below
thf(fact_8272_sum__bounded__above,axiom,
    ! [A3: set_real,F: real > real,K4: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8273_sum__bounded__above,axiom,
    ! [A3: set_o,F: $o > real,K4: real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8274_sum__bounded__above,axiom,
    ! [A3: set_complex,F: complex > real,K4: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8275_sum__bounded__above,axiom,
    ! [A3: set_int,F: int > real,K4: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8276_sum__bounded__above,axiom,
    ! [A3: set_real,F: real > rat,K4: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8277_sum__bounded__above,axiom,
    ! [A3: set_o,F: $o > rat,K4: rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8278_sum__bounded__above,axiom,
    ! [A3: set_complex,F: complex > rat,K4: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8279_sum__bounded__above,axiom,
    ! [A3: set_nat,F: nat > rat,K4: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8280_sum__bounded__above,axiom,
    ! [A3: set_int,F: int > rat,K4: rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8281_sum__bounded__above,axiom,
    ! [A3: set_real,F: real > nat,K4: nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ K4 ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).

% sum_bounded_above
thf(fact_8282_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: real > real,H2: real > real] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A3 @ C2 )
       => ( ( ord_less_eq_set_real @ B2 @ C2 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G3 @ A3 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) )
                = ( ( groups8097168146408367636l_real @ G3 @ C2 )
                  = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8283_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A3 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8691415230153176458o_real @ G3 @ A3 )
                  = ( groups8691415230153176458o_real @ H2 @ B2 ) )
                = ( ( groups8691415230153176458o_real @ G3 @ C2 )
                  = ( groups8691415230153176458o_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8284_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G3 @ A3 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) )
                = ( ( groups5808333547571424918x_real @ G3 @ C2 )
                  = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8285_sum_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A: extended_enat] :
                ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: extended_enat] :
                  ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G3 @ A3 )
                  = ( groups4148127829035722712t_real @ H2 @ B2 ) )
                = ( ( groups4148127829035722712t_real @ G3 @ C2 )
                  = ( groups4148127829035722712t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8286_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A3 @ C2 )
       => ( ( ord_less_eq_set_real @ B2 @ C2 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G3 @ A3 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
                = ( ( groups1300246762558778688al_rat @ G3 @ C2 )
                  = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8287_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > rat,H2: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A3 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
                  = ( groups7872700643590313910_o_rat @ H2 @ B2 ) )
                = ( ( groups7872700643590313910_o_rat @ G3 @ C2 )
                  = ( groups7872700643590313910_o_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8288_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
                = ( ( groups5058264527183730370ex_rat @ G3 @ C2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8289_sum_Osame__carrier,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A: extended_enat] :
                ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: extended_enat] :
                  ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups1392844769737527556at_rat @ G3 @ A3 )
                  = ( groups1392844769737527556at_rat @ H2 @ B2 ) )
                = ( ( groups1392844769737527556at_rat @ G3 @ C2 )
                  = ( groups1392844769737527556at_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8290_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A3 @ C2 )
       => ( ( ord_less_eq_set_real @ B2 @ C2 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_nat ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G3 @ A3 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
                = ( ( groups1935376822645274424al_nat @ G3 @ C2 )
                  = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8291_sum_Osame__carrier,axiom,
    ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A3 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_nat ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
                  = ( groups8507830703676809646_o_nat @ H2 @ B2 ) )
                = ( ( groups8507830703676809646_o_nat @ G3 @ C2 )
                  = ( groups8507830703676809646_o_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8292_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: real > real,H2: real > real] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A3 @ C2 )
       => ( ( ord_less_eq_set_real @ B2 @ C2 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G3 @ C2 )
                  = ( groups8097168146408367636l_real @ H2 @ C2 ) )
               => ( ( groups8097168146408367636l_real @ G3 @ A3 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8293_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A3 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8691415230153176458o_real @ G3 @ C2 )
                  = ( groups8691415230153176458o_real @ H2 @ C2 ) )
               => ( ( groups8691415230153176458o_real @ G3 @ A3 )
                  = ( groups8691415230153176458o_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8294_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G3 @ C2 )
                  = ( groups5808333547571424918x_real @ H2 @ C2 ) )
               => ( ( groups5808333547571424918x_real @ G3 @ A3 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8295_sum_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A: extended_enat] :
                ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_real ) )
           => ( ! [B: extended_enat] :
                  ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G3 @ C2 )
                  = ( groups4148127829035722712t_real @ H2 @ C2 ) )
               => ( ( groups4148127829035722712t_real @ G3 @ A3 )
                  = ( groups4148127829035722712t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8296_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A3 @ C2 )
       => ( ( ord_less_eq_set_real @ B2 @ C2 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G3 @ C2 )
                  = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
               => ( ( groups1300246762558778688al_rat @ G3 @ A3 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8297_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > rat,H2: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A3 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G3 @ C2 )
                  = ( groups7872700643590313910_o_rat @ H2 @ C2 ) )
               => ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
                  = ( groups7872700643590313910_o_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8298_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G3 @ C2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
               => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8299_sum_Osame__carrierI,axiom,
    ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ C2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
         => ( ! [A: extended_enat] :
                ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_rat ) )
           => ( ! [B: extended_enat] :
                  ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups1392844769737527556at_rat @ G3 @ C2 )
                  = ( groups1392844769737527556at_rat @ H2 @ C2 ) )
               => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
                  = ( groups1392844769737527556at_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8300_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B2: set_real,G3: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A3 @ C2 )
       => ( ( ord_less_eq_set_real @ B2 @ C2 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_nat ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G3 @ C2 )
                  = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
               => ( ( groups1935376822645274424al_nat @ G3 @ A3 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8301_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A3 @ C2 )
       => ( ( ord_less_eq_set_o @ B2 @ C2 )
         => ( ! [A: $o] :
                ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
               => ( ( G3 @ A )
                  = zero_zero_nat ) )
           => ( ! [B: $o] :
                  ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
                 => ( ( H2 @ B )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G3 @ C2 )
                  = ( groups8507830703676809646_o_nat @ H2 @ C2 ) )
               => ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
                  = ( groups8507830703676809646_o_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8302_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G3 @ S )
            = ( groups5808333547571424918x_real @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8303_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G3 @ S )
            = ( groups4148127829035722712t_real @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8304_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G3 @ S )
            = ( groups5058264527183730370ex_rat @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8305_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups1392844769737527556at_rat @ G3 @ S )
            = ( groups1392844769737527556at_rat @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8306_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G3 @ S )
            = ( groups5693394587270226106ex_nat @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8307_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G3 @ S )
            = ( groups2027974829824023292at_nat @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8308_sum_Omono__neutral__left,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G3 @ S )
            = ( groups5690904116761175830ex_int @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8309_sum_Omono__neutral__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups2025484359314973016at_int @ G3 @ S )
            = ( groups2025484359314973016at_int @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8310_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G3: nat > rat] :
      ( ( finite_finite_nat @ T2 )
     => ( ( ord_less_eq_set_nat @ S @ T2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G3 @ S )
            = ( groups2906978787729119204at_rat @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8311_sum_Omono__neutral__left,axiom,
    ! [T2: set_nat,S: set_nat,G3: nat > int] :
      ( ( finite_finite_nat @ T2 )
     => ( ( ord_less_eq_set_nat @ S @ T2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G3 @ S )
            = ( groups3539618377306564664at_int @ G3 @ T2 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8312_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G3 @ T2 )
            = ( groups5808333547571424918x_real @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8313_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G3 @ T2 )
            = ( groups4148127829035722712t_real @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8314_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G3 @ T2 )
            = ( groups5058264527183730370ex_rat @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8315_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups1392844769737527556at_rat @ G3 @ T2 )
            = ( groups1392844769737527556at_rat @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8316_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G3 @ T2 )
            = ( groups5693394587270226106ex_nat @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8317_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G3 @ T2 )
            = ( groups2027974829824023292at_nat @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8318_sum_Omono__neutral__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G3 @ T2 )
            = ( groups5690904116761175830ex_int @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8319_sum_Omono__neutral__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups2025484359314973016at_int @ G3 @ T2 )
            = ( groups2025484359314973016at_int @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8320_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G3: nat > rat] :
      ( ( finite_finite_nat @ T2 )
     => ( ( ord_less_eq_set_nat @ S @ T2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G3 @ T2 )
            = ( groups2906978787729119204at_rat @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8321_sum_Omono__neutral__right,axiom,
    ! [T2: set_nat,S: set_nat,G3: nat > int] :
      ( ( finite_finite_nat @ T2 )
     => ( ( ord_less_eq_set_nat @ S @ T2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G3 @ T2 )
            = ( groups3539618377306564664at_int @ G3 @ S ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8322_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S: set_real,H2: real > real,G3: real > real] :
      ( ( finite_finite_real @ T2 )
     => ( ( ord_less_eq_set_real @ S @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G3 @ S )
              = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8323_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S: set_o,H2: $o > real,G3: $o > real] :
      ( ( finite_finite_o @ T2 )
     => ( ( ord_less_eq_set_o @ S @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8691415230153176458o_real @ G3 @ S )
              = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8324_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S: set_complex,H2: complex > real,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G3 @ S )
              = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8325_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4148127829035722712t_real @ G3 @ S )
              = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8326_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S: set_real,H2: real > rat,G3: real > rat] :
      ( ( finite_finite_real @ T2 )
     => ( ( ord_less_eq_set_real @ S @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G3 @ S )
              = ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8327_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S: set_o,H2: $o > rat,G3: $o > rat] :
      ( ( finite_finite_o @ T2 )
     => ( ( ord_less_eq_set_o @ S @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G3 @ S )
              = ( groups7872700643590313910_o_rat @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8328_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_complex,S: set_complex,H2: complex > rat,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G3 @ S )
              = ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8329_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1392844769737527556at_rat @ G3 @ S )
              = ( groups1392844769737527556at_rat @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8330_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_real,S: set_real,H2: real > nat,G3: real > nat] :
      ( ( finite_finite_real @ T2 )
     => ( ( ord_less_eq_set_real @ S @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G3 @ S )
              = ( groups1935376822645274424al_nat @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8331_sum_Omono__neutral__cong__left,axiom,
    ! [T2: set_o,S: set_o,H2: $o > nat,G3: $o > nat] :
      ( ( finite_finite_o @ T2 )
     => ( ( ord_less_eq_set_o @ S @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G3 @ S )
              = ( groups8507830703676809646_o_nat @ H2 @ T2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8332_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S: set_real,G3: real > real,H2: real > real] :
      ( ( finite_finite_real @ T2 )
     => ( ( ord_less_eq_set_real @ S @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G3 @ T2 )
              = ( groups8097168146408367636l_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8333_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S: set_o,G3: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ T2 )
     => ( ( ord_less_eq_set_o @ S @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8691415230153176458o_real @ G3 @ T2 )
              = ( groups8691415230153176458o_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8334_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G3 @ T2 )
              = ( groups5808333547571424918x_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8335_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4148127829035722712t_real @ G3 @ T2 )
              = ( groups4148127829035722712t_real @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8336_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S: set_real,G3: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T2 )
     => ( ( ord_less_eq_set_real @ S @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G3 @ T2 )
              = ( groups1300246762558778688al_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8337_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S: set_o,G3: $o > rat,H2: $o > rat] :
      ( ( finite_finite_o @ T2 )
     => ( ( ord_less_eq_set_o @ S @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G3 @ T2 )
              = ( groups7872700643590313910_o_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8338_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_complex,S: set_complex,G3: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( ord_le211207098394363844omplex @ S @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G3 @ T2 )
              = ( groups5058264527183730370ex_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8339_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1392844769737527556at_rat @ G3 @ T2 )
              = ( groups1392844769737527556at_rat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8340_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_real,S: set_real,G3: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T2 )
     => ( ( ord_less_eq_set_real @ S @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G3 @ T2 )
              = ( groups1935376822645274424al_nat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8341_sum_Omono__neutral__cong__right,axiom,
    ! [T2: set_o,S: set_o,G3: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ T2 )
     => ( ( ord_less_eq_set_o @ S @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S )
               => ( ( G3 @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G3 @ T2 )
              = ( groups8507830703676809646_o_nat @ H2 @ S ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8342_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5808333547571424918x_real @ G3 @ A3 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8343_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups4148127829035722712t_real @ G3 @ A3 )
          = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8344_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8345_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > rat] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
          = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8346_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8347_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
          = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8348_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A3: set_complex,G3: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A3 )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8349_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > int] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( groups2025484359314973016at_int @ G3 @ A3 )
          = ( plus_plus_int @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2025484359314973016at_int @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8350_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A3: set_nat,G3: nat > rat] :
      ( ( ord_less_eq_set_nat @ B2 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups2906978787729119204at_rat @ G3 @ A3 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8351_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A3: set_nat,G3: nat > int] :
      ( ( ord_less_eq_set_nat @ B2 @ A3 )
     => ( ( finite_finite_nat @ A3 )
       => ( ( groups3539618377306564664at_int @ G3 @ A3 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( groups3539618377306564664at_int @ G3 @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8352_sum__diff,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8353_sum__diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
       => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
          = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8354_sum__diff,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8355_sum__diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
       => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8356_sum__diff,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8357_sum__diff,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
       => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
          = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A3 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8358_sum__diff,axiom,
    ! [A3: set_nat,B2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B2 @ A3 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8359_sum__diff,axiom,
    ! [A3: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B2 @ A3 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8360_sum__diff,axiom,
    ! [A3: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_int @ B2 @ A3 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8361_sum__diff,axiom,
    ! [A3: set_int,B2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ord_less_eq_set_int @ B2 @ A3 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_8362_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S: set_real,H2: real > real,G3: real > real] :
      ( ( finite_finite_real @ T2 )
     => ( ( finite_finite_real @ S )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ ( minus_minus_set_real @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ ( inf_inf_set_real @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups8097168146408367636l_real @ G3 @ S )
                = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8363_sum_Omono__neutral__cong,axiom,
    ! [T2: set_o,S: set_o,H2: $o > real,G3: $o > real] :
      ( ( finite_finite_o @ T2 )
     => ( ( finite_finite_o @ S )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ ( minus_minus_set_o @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ ( inf_inf_set_o @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups8691415230153176458o_real @ G3 @ S )
                = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8364_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,H2: int > real,G3: int > real] :
      ( ( finite_finite_int @ T2 )
     => ( ( finite_finite_int @ S )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ ( minus_minus_set_int @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ ( inf_inf_set_int @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups8778361861064173332t_real @ G3 @ S )
                = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8365_sum_Omono__neutral__cong,axiom,
    ! [T2: set_complex,S: set_complex,H2: complex > real,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( finite3207457112153483333omplex @ S )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups5808333547571424918x_real @ G3 @ S )
                = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8366_sum_Omono__neutral__cong,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( finite4001608067531595151d_enat @ S )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups4148127829035722712t_real @ G3 @ S )
                = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8367_sum_Omono__neutral__cong,axiom,
    ! [T2: set_real,S: set_real,H2: real > rat,G3: real > rat] :
      ( ( finite_finite_real @ T2 )
     => ( ( finite_finite_real @ S )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ ( minus_minus_set_real @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ ( minus_minus_set_real @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ ( inf_inf_set_real @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups1300246762558778688al_rat @ G3 @ S )
                = ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8368_sum_Omono__neutral__cong,axiom,
    ! [T2: set_o,S: set_o,H2: $o > rat,G3: $o > rat] :
      ( ( finite_finite_o @ T2 )
     => ( ( finite_finite_o @ S )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ ( minus_minus_set_o @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ ( minus_minus_set_o @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ ( inf_inf_set_o @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups7872700643590313910_o_rat @ G3 @ S )
                = ( groups7872700643590313910_o_rat @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8369_sum_Omono__neutral__cong,axiom,
    ! [T2: set_int,S: set_int,H2: int > rat,G3: int > rat] :
      ( ( finite_finite_int @ T2 )
     => ( ( finite_finite_int @ S )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ ( minus_minus_set_int @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ ( inf_inf_set_int @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups3906332499630173760nt_rat @ G3 @ S )
                = ( groups3906332499630173760nt_rat @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8370_sum_Omono__neutral__cong,axiom,
    ! [T2: set_complex,S: set_complex,H2: complex > rat,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ T2 )
     => ( ( finite3207457112153483333omplex @ S )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups5058264527183730370ex_rat @ G3 @ S )
                = ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8371_sum_Omono__neutral__cong,axiom,
    ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ T2 )
     => ( ( finite4001608067531595151d_enat @ S )
       => ( ! [I2: extended_enat] :
              ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: extended_enat] :
                ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ S @ T2 ) )
               => ( ( G3 @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ S @ T2 ) )
                 => ( ( G3 @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups1392844769737527556at_rat @ G3 @ S )
                = ( groups1392844769737527556at_rat @ H2 @ T2 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8372_sum_Ounion__inter,axiom,
    ! [A3: set_int,B2: set_int,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8373_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8374_sum_Ounion__inter,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
          = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8375_sum_Ounion__inter,axiom,
    ! [A3: set_int,B2: set_int,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8376_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8377_sum_Ounion__inter,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
          = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8378_sum_Ounion__inter,axiom,
    ! [A3: set_int,B2: set_int,G3: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8379_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8380_sum_Ounion__inter,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
          = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8381_sum_Ounion__inter,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8382_sum_OInt__Diff,axiom,
    ! [A3: set_int,G3: int > real,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real @ G3 @ A3 )
        = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8383_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: complex > real,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G3 @ A3 )
        = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8384_sum_OInt__Diff,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > real,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G3 @ A3 )
        = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8385_sum_OInt__Diff,axiom,
    ! [A3: set_int,G3: int > rat,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8386_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: complex > rat,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
        = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8387_sum_OInt__Diff,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > rat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
        = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8388_sum_OInt__Diff,axiom,
    ! [A3: set_int,G3: int > nat,B2: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8389_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: complex > nat,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
        = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8390_sum_OInt__Diff,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > nat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
        = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8391_sum_OInt__Diff,axiom,
    ! [A3: set_complex,G3: complex > int,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
        = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8392_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A2: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
        @ A2 )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A2 @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8393_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A2: complex] :
      ( ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
        @ A2 )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A2 @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8394_sums__Suc__imp,axiom,
    ! [F: nat > real,S2: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
          @ S2 )
       => ( sums_real @ F @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_8395_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8396_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8397_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8398_sums__Suc__iff,axiom,
    ! [F: nat > real,S2: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ S2 )
      = ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8399_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > real,S2: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ( F @ I2 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
          @ S2 )
        = ( sums_real @ F @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_8400_sum_OIf__cases,axiom,
    ! [A3: set_real,P: real > $o,H2: real > real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_real @ ( groups8097168146408367636l_real @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups8097168146408367636l_real @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8401_sum_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H2: int > real,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_real @ ( groups8778361861064173332t_real @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8402_sum_OIf__cases,axiom,
    ! [A3: set_complex,P: complex > $o,H2: complex > real,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_real @ ( groups5808333547571424918x_real @ H2 @ ( inf_inf_set_complex @ A3 @ ( collect_complex @ P ) ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8403_sum_OIf__cases,axiom,
    ! [A3: set_Extended_enat,P: extended_enat > $o,H2: extended_enat > real,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real
          @ ^ [X: extended_enat] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_real @ ( groups4148127829035722712t_real @ H2 @ ( inf_in8357106775501769908d_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ ( uminus417252749190364093d_enat @ ( collec4429806609662206161d_enat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8404_sum_OIf__cases,axiom,
    ! [A3: set_real,P: real > $o,H2: real > rat,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat
          @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups1300246762558778688al_rat @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8405_sum_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H2: int > rat,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8406_sum_OIf__cases,axiom,
    ! [A3: set_complex,P: complex > $o,H2: complex > rat,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ H2 @ ( inf_inf_set_complex @ A3 @ ( collect_complex @ P ) ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8407_sum_OIf__cases,axiom,
    ! [A3: set_Extended_enat,P: extended_enat > $o,H2: extended_enat > rat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat
          @ ^ [X: extended_enat] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ H2 @ ( inf_in8357106775501769908d_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ ( uminus417252749190364093d_enat @ ( collec4429806609662206161d_enat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8408_sum_OIf__cases,axiom,
    ! [A3: set_real,P: real > $o,H2: real > nat,G3: real > nat] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( if_nat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups1935376822645274424al_nat @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups1935376822645274424al_nat @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8409_sum_OIf__cases,axiom,
    ! [A3: set_int,P: int > $o,H2: int > nat,G3: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( if_nat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8410_sum__power__add,axiom,
    ! [X2: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8411_sum__power__add,axiom,
    ! [X2: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8412_sum__power__add,axiom,
    ! [X2: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8413_sum__power__add,axiom,
    ! [X2: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8414_sum__mono2,axiom,
    ! [B2: set_real,A3: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A3 @ B2 )
       => ( ! [B: real] :
              ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8415_sum__mono2,axiom,
    ! [B2: set_o,A3: set_o,F: $o > real] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A3 @ B2 )
       => ( ! [B: $o] :
              ( ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8416_sum__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ! [B: complex] :
              ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8417_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ! [B: extended_enat] :
              ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8418_sum__mono2,axiom,
    ! [B2: set_real,A3: set_real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A3 @ B2 )
       => ( ! [B: real] :
              ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8419_sum__mono2,axiom,
    ! [B2: set_o,A3: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A3 @ B2 )
       => ( ! [B: $o] :
              ( ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8420_sum__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ! [B: complex] :
              ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8421_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ! [B: extended_enat] :
              ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8422_sum__mono2,axiom,
    ! [B2: set_nat,A3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( ! [B: nat] :
              ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A3 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8423_sum__mono2,axiom,
    ! [B2: set_real,A3: set_real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A3 @ B2 )
       => ( ! [B: real] :
              ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8424_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B2: set_int,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8425_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8426_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8427_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B2: set_int,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8428_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8429_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_rat ) )
         => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8430_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_int,B2: set_int,G3: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8431_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8432_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8433_sum_Ounion__inter__neutral,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
             => ( ( G3 @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8434_sum_OatLeastAtMost__rev,axiom,
    ! [G3: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8435_sum_OatLeastAtMost__rev,axiom,
    ! [G3: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8436_sum_Oremove,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( groups5808333547571424918x_real @ G3 @ A3 )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8437_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( groups4148127829035722712t_real @ G3 @ A3 )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8438_sum_Oremove,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8439_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8440_sum_Oremove,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
          = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8441_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
          = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8442_sum_Oremove,axiom,
    ! [A3: set_complex,X2: complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( member_complex @ X2 @ A3 )
       => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
          = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8443_sum_Oremove,axiom,
    ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( member_Extended_enat @ X2 @ A3 )
       => ( ( groups2025484359314973016at_int @ G3 @ A3 )
          = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8444_sum_Oremove,axiom,
    ! [A3: set_real,X2: real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ( groups8097168146408367636l_real @ G3 @ A3 )
          = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8445_sum_Oremove,axiom,
    ! [A3: set_real,X2: real,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( member_real @ X2 @ A3 )
       => ( ( groups1300246762558778688al_rat @ G3 @ A3 )
          = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8446_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: complex > real,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
        = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8447_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > real,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
        = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8448_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: complex > rat,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
        = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8449_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > rat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups1392844769737527556at_rat @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
        = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8450_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: complex > nat,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5693394587270226106ex_nat @ G3 @ ( insert_complex @ X2 @ A3 ) )
        = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8451_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > nat,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups2027974829824023292at_nat @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
        = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8452_sum_Oinsert__remove,axiom,
    ! [A3: set_complex,G3: complex > int,X2: complex] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( groups5690904116761175830ex_int @ G3 @ ( insert_complex @ X2 @ A3 ) )
        = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8453_sum_Oinsert__remove,axiom,
    ! [A3: set_Extended_enat,G3: extended_enat > int,X2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( groups2025484359314973016at_int @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
        = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8454_sum_Oinsert__remove,axiom,
    ! [A3: set_real,G3: real > real,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
        = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8455_sum_Oinsert__remove,axiom,
    ! [A3: set_real,G3: real > rat,X2: real] :
      ( ( finite_finite_real @ A3 )
     => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
        = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8456_sum__diff1,axiom,
    ! [A3: set_complex,A2: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( member_complex @ A2 @ A3 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_complex @ A2 @ A3 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8457_sum__diff1,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( member_Extended_enat @ A2 @ A3 )
         => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ A3 )
         => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups4148127829035722712t_real @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8458_sum__diff1,axiom,
    ! [A3: set_real,A2: real,F: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( member_real @ A2 @ A3 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_real @ A2 @ A3 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8459_sum__diff1,axiom,
    ! [A3: set_o,A2: $o,F: $o > real] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( member_o @ A2 @ A3 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
            = ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_o @ A2 @ A3 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
            = ( groups8691415230153176458o_real @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8460_sum__diff1,axiom,
    ! [A3: set_int,A2: int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( member_int @ A2 @ A3 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_int @ A2 @ A3 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8461_sum__diff1,axiom,
    ! [A3: set_complex,A2: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( member_complex @ A2 @ A3 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_complex @ A2 @ A3 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8462_sum__diff1,axiom,
    ! [A3: set_Extended_enat,A2: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( member_Extended_enat @ A2 @ A3 )
         => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ A3 )
         => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups1392844769737527556at_rat @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8463_sum__diff1,axiom,
    ! [A3: set_real,A2: real,F: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( ( member_real @ A2 @ A3 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_real @ A2 @ A3 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8464_sum__diff1,axiom,
    ! [A3: set_o,A2: $o,F: $o > rat] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( member_o @ A2 @ A3 )
         => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
            = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_o @ A2 @ A3 )
         => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
            = ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8465_sum__diff1,axiom,
    ! [A3: set_int,A2: int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( member_int @ A2 @ A3 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
        & ( ~ ( member_int @ A2 @ A3 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ) ) ).

% sum_diff1
thf(fact_8466_sum__Un,axiom,
    ! [A3: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8467_sum__Un,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8468_sum__Un,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups4148127829035722712t_real @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) @ ( groups4148127829035722712t_real @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8469_sum__Un,axiom,
    ! [A3: set_int,B2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8470_sum__Un,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8471_sum__Un,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups1392844769737527556at_rat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) @ ( groups1392844769737527556at_rat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8472_sum__Un,axiom,
    ! [A3: set_nat,B2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8473_sum__Un,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8474_sum__Un,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups2025484359314973016at_int @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups2025484359314973016at_int @ F @ A3 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) @ ( groups2025484359314973016at_int @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8475_sum__Un,axiom,
    ! [A3: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8476_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A3 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8477_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
            = bot_bo7653980558646680370d_enat )
         => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8478_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A3 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8479_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
            = bot_bo7653980558646680370d_enat )
         => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8480_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A3 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8481_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
            = bot_bo7653980558646680370d_enat )
         => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8482_sum_Ounion__disjoint,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A3 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8483_sum_Ounion__disjoint,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
            = bot_bo7653980558646680370d_enat )
         => ( ( groups2025484359314973016at_int @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
            = ( plus_plus_int @ ( groups2025484359314973016at_int @ G3 @ A3 ) @ ( groups2025484359314973016at_int @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8484_sum_Ounion__disjoint,axiom,
    ! [A3: set_real,B2: set_real,G3: real > real] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A3 @ B2 )
            = bot_bot_set_real )
         => ( ( groups8097168146408367636l_real @ G3 @ ( sup_sup_set_real @ A3 @ B2 ) )
            = ( plus_plus_real @ ( groups8097168146408367636l_real @ G3 @ A3 ) @ ( groups8097168146408367636l_real @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8485_sum_Ounion__disjoint,axiom,
    ! [A3: set_real,B2: set_real,G3: real > rat] :
      ( ( finite_finite_real @ A3 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A3 @ B2 )
            = bot_bot_set_real )
         => ( ( groups1300246762558778688al_rat @ G3 @ ( sup_sup_set_real @ A3 @ B2 ) )
            = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ G3 @ A3 ) @ ( groups1300246762558778688al_rat @ G3 @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8486_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B2: set_int,G3: int > real] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8487_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > real] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8488_sum_Ounion__diff2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8489_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B2: set_int,G3: int > rat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8490_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8491_sum_Ounion__diff2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8492_sum_Ounion__diff2,axiom,
    ! [A3: set_int,B2: set_int,G3: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8493_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8494_sum_Ounion__diff2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8495_sum_Ounion__diff2,axiom,
    ! [A3: set_complex,B2: set_complex,G3: complex > int] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8496_sum__Un2,axiom,
    ! [A3: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
     => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8497_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
     => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8498_sum__Un2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
     => ( ( groups4148127829035722712t_real @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups4148127829035722712t_real @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8499_sum__Un2,axiom,
    ! [A3: set_int,B2: set_int,F: int > rat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
     => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8500_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
     => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8501_sum__Un2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
     => ( ( groups1392844769737527556at_rat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups1392844769737527556at_rat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8502_sum__Un2,axiom,
    ! [A3: set_int,B2: set_int,F: int > nat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
     => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8503_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
     => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8504_sum__Un2,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
     => ( ( groups2027974829824023292at_nat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups2027974829824023292at_nat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8505_sum__Un2,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
     => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8506_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_8507_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_8508_suminf__finite,axiom,
    ! [N6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).

% suminf_finite
thf(fact_8509_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8510_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real,C: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ( member_Extended_enat @ A2 @ S )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ S )
         => ( ( groups4148127829035722712t_real
              @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8511_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8512_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > rat,C: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ( member_Extended_enat @ A2 @ S )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups1392844769737527556at_rat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ S )
         => ( ( groups1392844769737527556at_rat
              @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups1392844769737527556at_rat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8513_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_nat @ ( B3 @ A2 ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8514_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > nat,C: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ( member_Extended_enat @ A2 @ S )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K3: extended_enat] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_nat @ ( B3 @ A2 ) @ ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ S )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K3: extended_enat] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8515_sum_Odelta__remove,axiom,
    ! [S: set_complex,A2: complex,B3: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( ( member_complex @ A2 @ S )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_int @ ( B3 @ A2 ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A2 @ S )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8516_sum_Odelta__remove,axiom,
    ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > int,C: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ( member_Extended_enat @ A2 @ S )
         => ( ( groups2025484359314973016at_int
              @ ^ [K3: extended_enat] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_int @ ( B3 @ A2 ) @ ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A2 @ S )
         => ( ( groups2025484359314973016at_int
              @ ^ [K3: extended_enat] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8517_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B3: real > real,C: real > real] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A2 @ S )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A2 @ S )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8518_sum_Odelta__remove,axiom,
    ! [S: set_real,A2: real,B3: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( ( member_real @ A2 @ S )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A2 @ S )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S )
            = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8519_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8520_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8521_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8522_powser__sums__zero,axiom,
    ! [A2: nat > complex] :
      ( sums_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( A2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
      @ ( A2 @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8523_powser__sums__zero,axiom,
    ! [A2: nat > real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( A2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
      @ ( A2 @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8524_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8525_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8526_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8527_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8528_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8529_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8530_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8531_sum_OatLeast0__atMost__Suc,axiom,
    ! [G3: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8532_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G3 @ M ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8533_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_int @ ( G3 @ M ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8534_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G3 @ M ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8535_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G3: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_real @ ( G3 @ M ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8536_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G3 @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8537_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G3 @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8538_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G3 @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8539_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G3: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G3 @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8540_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_8541_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_8542_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_8543_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_8544_sum__strict__mono2,axiom,
    ! [B2: set_real,A3: set_real,B3: real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A3 @ B2 )
       => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8545_sum__strict__mono2,axiom,
    ! [B2: set_o,A3: set_o,B3: $o,F: $o > real] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A3 @ B2 )
       => ( ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8546_sum__strict__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,B3: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8547_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ( member_Extended_enat @ B3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8548_sum__strict__mono2,axiom,
    ! [B2: set_real,A3: set_real,B3: real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A3 @ B2 )
       => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8549_sum__strict__mono2,axiom,
    ! [B2: set_o,A3: set_o,B3: $o,F: $o > rat] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A3 @ B2 )
       => ( ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8550_sum__strict__mono2,axiom,
    ! [B2: set_complex,A3: set_complex,B3: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
       => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8551_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: extended_enat > rat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
       => ( ( member_Extended_enat @ B3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8552_sum__strict__mono2,axiom,
    ! [B2: set_nat,A3: set_nat,B3: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8553_sum__strict__mono2,axiom,
    ! [B2: set_real,A3: set_real,B3: real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A3 @ B2 )
       => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8554_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A3 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8555_member__le__sum,axiom,
    ! [I: extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
      ( ( member_Extended_enat @ I @ A3 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite4001608067531595151d_enat @ A3 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups4148127829035722712t_real @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8556_member__le__sum,axiom,
    ! [I: real,A3: set_real,F: real > real] :
      ( ( member_real @ I @ A3 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A3 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8557_member__le__sum,axiom,
    ! [I: $o,A3: set_o,F: $o > real] :
      ( ( member_o @ I @ A3 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ I @ bot_bot_set_o ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A3 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8558_member__le__sum,axiom,
    ! [I: int,A3: set_int,F: int > real] :
      ( ( member_int @ I @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A3 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8559_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A3 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A3 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8560_member__le__sum,axiom,
    ! [I: extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
      ( ( member_Extended_enat @ I @ A3 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite4001608067531595151d_enat @ A3 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1392844769737527556at_rat @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8561_member__le__sum,axiom,
    ! [I: real,A3: set_real,F: real > rat] :
      ( ( member_real @ I @ A3 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A3 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8562_member__le__sum,axiom,
    ! [I: $o,A3: set_o,F: $o > rat] :
      ( ( member_o @ I @ A3 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ I @ bot_bot_set_o ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A3 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8563_member__le__sum,axiom,
    ! [I: int,A3: set_int,F: int > rat] :
      ( ( member_int @ I @ A3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A3 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ) ) ).

% member_le_sum
thf(fact_8564_sum__bounded__above__strict,axiom,
    ! [A3: set_real,F: real > rat,K4: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
       => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8565_sum__bounded__above__strict,axiom,
    ! [A3: set_o,F: $o > rat,K4: rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
       => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8566_sum__bounded__above__strict,axiom,
    ! [A3: set_complex,F: complex > rat,K4: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
       => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8567_sum__bounded__above__strict,axiom,
    ! [A3: set_nat,F: nat > rat,K4: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
       => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8568_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: int > rat,K4: rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
       => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8569_sum__bounded__above__strict,axiom,
    ! [A3: set_real,F: real > nat,K4: nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
       => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8570_sum__bounded__above__strict,axiom,
    ! [A3: set_o,F: $o > nat,K4: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
       => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8571_sum__bounded__above__strict,axiom,
    ! [A3: set_complex,F: complex > nat,K4: nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
       => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8572_sum__bounded__above__strict,axiom,
    ! [A3: set_int,F: int > nat,K4: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
       => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8573_sum__bounded__above__strict,axiom,
    ! [A3: set_real,F: real > int,K4: int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
       => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_8574_sum__bounded__above__divide,axiom,
    ! [A3: set_complex,F: complex > real,K4: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) ) ) )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( A3 != bot_bot_set_complex )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8575_sum__bounded__above__divide,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > real,K4: real] :
      ( ! [I2: extended_enat] :
          ( ( member_Extended_enat @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite121521170596916366d_enat @ A3 ) ) ) ) )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8576_sum__bounded__above__divide,axiom,
    ! [A3: set_real,F: real > real,K4: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) ) ) )
     => ( ( finite_finite_real @ A3 )
       => ( ( A3 != bot_bot_set_real )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8577_sum__bounded__above__divide,axiom,
    ! [A3: set_o,F: $o > real,K4: real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) ) ) )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8578_sum__bounded__above__divide,axiom,
    ! [A3: set_int,F: int > real,K4: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A3 )
         => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) ) ) )
     => ( ( finite_finite_int @ A3 )
       => ( ( A3 != bot_bot_set_int )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8579_sum__bounded__above__divide,axiom,
    ! [A3: set_complex,F: complex > rat,K4: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) ) ) )
     => ( ( finite3207457112153483333omplex @ A3 )
       => ( ( A3 != bot_bot_set_complex )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8580_sum__bounded__above__divide,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > rat,K4: rat] :
      ( ! [I2: extended_enat] :
          ( ( member_Extended_enat @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite121521170596916366d_enat @ A3 ) ) ) ) )
     => ( ( finite4001608067531595151d_enat @ A3 )
       => ( ( A3 != bot_bo7653980558646680370d_enat )
         => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8581_sum__bounded__above__divide,axiom,
    ! [A3: set_real,F: real > rat,K4: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) ) ) )
     => ( ( finite_finite_real @ A3 )
       => ( ( A3 != bot_bot_set_real )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8582_sum__bounded__above__divide,axiom,
    ! [A3: set_o,F: $o > rat,K4: rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) ) ) )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8583_sum__bounded__above__divide,axiom,
    ! [A3: set_nat,F: nat > rat,K4: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A3 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) ) ) )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ K4 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_8584_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G3 @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8585_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G3 @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8586_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G3 @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8587_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G3: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G3 @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8588_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8589_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8590_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8591_convex__sum__bound__le,axiom,
    ! [I5: set_real,X2: real > code_integer,A2: real > code_integer,B3: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X2 @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8592_convex__sum__bound__le,axiom,
    ! [I5: set_o,X2: $o > code_integer,A2: $o > code_integer,B3: code_integer,Delta: code_integer] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
     => ( ( ( groups4406642042086082107nteger @ X2 @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups4406642042086082107nteger
                  @ ^ [I4: $o] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8593_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X2: nat > code_integer,A2: nat > code_integer,B3: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X2 @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8594_convex__sum__bound__le,axiom,
    ! [I5: set_int,X2: int > code_integer,A2: int > code_integer,B3: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X2 @ I5 )
          = one_one_Code_integer )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8595_convex__sum__bound__le,axiom,
    ! [I5: set_real,X2: real > real,A2: real > real,B3: real,Delta: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X2 @ I5 )
          = one_one_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I4: real] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8596_convex__sum__bound__le,axiom,
    ! [I5: set_o,X2: $o > real,A2: $o > real,B3: real,Delta: real] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
     => ( ( ( groups8691415230153176458o_real @ X2 @ I5 )
          = one_one_real )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8691415230153176458o_real
                  @ ^ [I4: $o] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8597_convex__sum__bound__le,axiom,
    ! [I5: set_int,X2: int > real,A2: int > real,B3: real,Delta: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X2 @ I5 )
          = one_one_real )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I4: int] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8598_convex__sum__bound__le,axiom,
    ! [I5: set_real,X2: real > rat,A2: real > rat,B3: rat,Delta: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X2 @ I5 )
          = one_one_rat )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I4: real] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8599_convex__sum__bound__le,axiom,
    ! [I5: set_o,X2: $o > rat,A2: $o > rat,B3: rat,Delta: rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
     => ( ( ( groups7872700643590313910_o_rat @ X2 @ I5 )
          = one_one_rat )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups7872700643590313910_o_rat
                  @ ^ [I4: $o] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8600_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X2: nat > rat,A2: nat > rat,B3: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X2 @ I5 )
          = one_one_rat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I4: nat] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8601_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_8602_sum__norm__bound,axiom,
    ! [S: set_real,F: real > complex,K4: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8603_sum__norm__bound,axiom,
    ! [S: set_o,F: $o > complex,K4: real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8604_sum__norm__bound,axiom,
    ! [S: set_list_nat,F: list_nat > complex,K4: real] :
      ( ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6529277132148336714omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8605_sum__norm__bound,axiom,
    ! [S: set_set_nat,F: set_nat > complex,K4: real] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_nat @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8606_sum__norm__bound,axiom,
    ! [S: set_nat,F: nat > complex,K4: real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8607_sum__norm__bound,axiom,
    ! [S: set_int,F: int > complex,K4: real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8608_sum__norm__bound,axiom,
    ! [S: set_complex,F: complex > complex,K4: real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8609_sum__norm__bound,axiom,
    ! [S: set_nat,F: nat > real,K4: real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ K4 ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S ) ) @ K4 ) ) ) ).

% sum_norm_bound
thf(fact_8610_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: nat > rat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8611_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: nat > int,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8612_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: nat > nat,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8613_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G3: nat > real,P6: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8614_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8615_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8616_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8617_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_8618_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_8619_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_8620_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8621_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8622_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8623_card__3__iff,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( ( S
              = ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ Z2 @ bot_bo2099793752762293965at_nat ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8624_card__3__iff,axiom,
    ! [S: set_complex] :
      ( ( ( finite_card_complex @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: complex,Y: complex,Z2: complex] :
            ( ( S
              = ( insert_complex @ X @ ( insert_complex @ Y @ ( insert_complex @ Z2 @ bot_bot_set_complex ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8625_card__3__iff,axiom,
    ! [S: set_list_nat] :
      ( ( ( finite_card_list_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: list_nat,Y: list_nat,Z2: list_nat] :
            ( ( S
              = ( insert_list_nat @ X @ ( insert_list_nat @ Y @ ( insert_list_nat @ Z2 @ bot_bot_set_list_nat ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8626_card__3__iff,axiom,
    ! [S: set_set_nat] :
      ( ( ( finite_card_set_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: set_nat,Y: set_nat,Z2: set_nat] :
            ( ( S
              = ( insert_set_nat @ X @ ( insert_set_nat @ Y @ ( insert_set_nat @ Z2 @ bot_bot_set_set_nat ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8627_card__3__iff,axiom,
    ! [S: set_real] :
      ( ( ( finite_card_real @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: real,Y: real,Z2: real] :
            ( ( S
              = ( insert_real @ X @ ( insert_real @ Y @ ( insert_real @ Z2 @ bot_bot_set_real ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8628_card__3__iff,axiom,
    ! [S: set_o] :
      ( ( ( finite_card_o @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: $o,Y: $o,Z2: $o] :
            ( ( S
              = ( insert_o @ X @ ( insert_o @ Y @ ( insert_o @ Z2 @ bot_bot_set_o ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8629_card__3__iff,axiom,
    ! [S: set_nat] :
      ( ( ( finite_card_nat @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: nat,Y: nat,Z2: nat] :
            ( ( S
              = ( insert_nat @ X @ ( insert_nat @ Y @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8630_card__3__iff,axiom,
    ! [S: set_int] :
      ( ( ( finite_card_int @ S )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X: int,Y: int,Z2: int] :
            ( ( S
              = ( insert_int @ X @ ( insert_int @ Y @ ( insert_int @ Z2 @ bot_bot_set_int ) ) ) )
            & ( X != Y )
            & ( Y != Z2 )
            & ( X != Z2 ) ) ) ) ).

% card_3_iff
thf(fact_8631_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_8632_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_8633_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_8634_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_8635_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_8636_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8637_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8638_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8639_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E2: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M3: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M3 )
                 => ! [N5: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8640_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E2: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M3: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M3 )
                 => ! [N5: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8641_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_8642_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_8643_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8644_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer )
      = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8645_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8646_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8647_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8648_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8649_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X2: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8650_sum_Oin__pairs,axiom,
    ! [G3: nat > rat,M: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8651_sum_Oin__pairs,axiom,
    ! [G3: nat > int,M: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8652_sum_Oin__pairs,axiom,
    ! [G3: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8653_sum_Oin__pairs,axiom,
    ! [G3: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( plus_plus_real @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8654_accp__subset__induct,axiom,
    ! [D4: produc4953844613479565601on_nat > $o,R: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,X2: produc4953844613479565601on_nat,P: produc4953844613479565601on_nat > $o] :
      ( ( ord_le8126618931240741628_nat_o @ D4 @ ( accp_P8646395344606611882on_nat @ R ) )
     => ( ! [X5: produc4953844613479565601on_nat,Z4: produc4953844613479565601on_nat] :
            ( ( D4 @ X5 )
           => ( ( R @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X2 )
         => ( ! [X5: produc4953844613479565601on_nat] :
                ( ( D4 @ X5 )
               => ( ! [Z5: produc4953844613479565601on_nat] :
                      ( ( R @ Z5 @ X5 )
                     => ( P @ Z5 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_8655_accp__subset__induct,axiom,
    ! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
     => ( ! [X5: product_prod_nat_nat,Z4: product_prod_nat_nat] :
            ( ( D4 @ X5 )
           => ( ( R @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X2 )
         => ( ! [X5: product_prod_nat_nat] :
                ( ( D4 @ X5 )
               => ( ! [Z5: product_prod_nat_nat] :
                      ( ( R @ Z5 @ X5 )
                     => ( P @ Z5 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_8656_accp__subset__induct,axiom,
    ! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
     => ( ! [X5: product_prod_int_int,Z4: product_prod_int_int] :
            ( ( D4 @ X5 )
           => ( ( R @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X2 )
         => ( ! [X5: product_prod_int_int] :
                ( ( D4 @ X5 )
               => ( ! [Z5: product_prod_int_int] :
                      ( ( R @ Z5 @ X5 )
                     => ( P @ Z5 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_8657_accp__subset__induct,axiom,
    ! [D4: list_nat > $o,R: list_nat > list_nat > $o,X2: list_nat,P: list_nat > $o] :
      ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
     => ( ! [X5: list_nat,Z4: list_nat] :
            ( ( D4 @ X5 )
           => ( ( R @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X2 )
         => ( ! [X5: list_nat] :
                ( ( D4 @ X5 )
               => ( ! [Z5: list_nat] :
                      ( ( R @ Z5 @ X5 )
                     => ( P @ Z5 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_8658_accp__subset__induct,axiom,
    ! [D4: nat > $o,R: nat > nat > $o,X2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
     => ( ! [X5: nat,Z4: nat] :
            ( ( D4 @ X5 )
           => ( ( R @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X2 )
         => ( ! [X5: nat] :
                ( ( D4 @ X5 )
               => ( ! [Z5: nat] :
                      ( ( R @ Z5 @ X5 )
                     => ( P @ Z5 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X2 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_8659_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8660_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_8661_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8662_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8663_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8664_double__arith__series,axiom,
    ! [A2: complex,D: complex,N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( plus_plus_complex @ A2 @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8665_double__arith__series,axiom,
    ! [A2: rat,D: rat,N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A2 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8666_double__arith__series,axiom,
    ! [A2: extended_enat,D: extended_enat,N: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I4: nat] : ( plus_p3455044024723400733d_enat @ A2 @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I4 ) @ 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 ) ) @ A2 ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8667_double__arith__series,axiom,
    ! [A2: code_integer,D: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ 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 ) ) @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8668_double__arith__series,axiom,
    ! [A2: int,D: int,N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8669_double__arith__series,axiom,
    ! [A2: nat,D: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8670_double__arith__series,axiom,
    ! [A2: real,D: real,N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( plus_plus_real @ A2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8671_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_8672_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_8673_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_8674_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_8675_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_8676_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_8677_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_8678_arith__series__nat,axiom,
    ! [A2: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8679_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8680_arith__series,axiom,
    ! [A2: code_integer,D: code_integer,N: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ 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 ) ) @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8681_arith__series,axiom,
    ! [A2: int,D: int,N: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8682_arith__series,axiom,
    ! [A2: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8683_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_8684_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_8685_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_8686_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_8687_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_8688_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_8689_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_8690_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_8691_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_8692_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_8693_sum__gp__offset,axiom,
    ! [X2: rat,M: nat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8694_sum__gp__offset,axiom,
    ! [X2: complex,M: nat,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X2 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8695_sum__gp__offset,axiom,
    ! [X2: real,M: nat,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8696_log__base__10__eq2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% log_base_10_eq2
thf(fact_8697_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_8698_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_8699_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_8700_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8701_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8702_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8703_and__int_Opinduct,axiom,
    ! [A0: int,A13: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A13 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A13 ) ) ) ).

% and_int.pinduct
thf(fact_8704_prod__decode__aux_Opelims,axiom,
    ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_8705_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > nat > nat,A13: nat,A24: nat,A33: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A13 @ ( product_Pair_nat_nat @ A24 @ A33 ) ) ) )
     => ( ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
            ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B @ A )
               => ( P @ F3 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F3 @ A @ Acc ) ) )
             => ( P @ F3 @ A @ B @ Acc ) ) )
       => ( P @ A0 @ A13 @ A24 @ A33 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_8706_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
        = Y3 )
     => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ Xa2 @ Xb ) ) )
       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
           => ( ( Y3 = none_P5556105721700978146at_nat )
             => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ V2 ) )
               => ( ( Xb = none_P5556105721700978146at_nat )
                 => ( ( Y3 = none_P5556105721700978146at_nat )
                   => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
           => ~ ! [A: product_prod_nat_nat] :
                  ( ( Xa2
                    = ( some_P7363390416028606310at_nat @ A ) )
                 => ! [B: product_prod_nat_nat] :
                      ( ( Xb
                        = ( some_P7363390416028606310at_nat @ B ) )
                     => ( ( Y3
                          = ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) )
                       => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_8707_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
        = Y3 )
     => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ Xa2 @ Xb ) ) )
       => ( ( ( Xa2 = none_num )
           => ( ( Y3 = none_num )
             => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
         => ( ! [V2: num] :
                ( ( Xa2
                  = ( some_num @ V2 ) )
               => ( ( Xb = none_num )
                 => ( ( Y3 = none_num )
                   => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
           => ~ ! [A: num] :
                  ( ( Xa2
                    = ( some_num @ A ) )
                 => ! [B: num] :
                      ( ( Xb
                        = ( some_num @ B ) )
                     => ( ( Y3
                          = ( some_num @ ( X2 @ A @ B ) ) )
                       => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_8708_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
        = Y3 )
     => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb ) ) )
       => ( ( ( Xa2 = none_nat )
           => ( ( Y3 = none_nat )
             => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
         => ( ! [V2: nat] :
                ( ( Xa2
                  = ( some_nat @ V2 ) )
               => ( ( Xb = none_nat )
                 => ( ( Y3 = none_nat )
                   => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
           => ~ ! [A: nat] :
                  ( ( Xa2
                    = ( some_nat @ A ) )
                 => ! [B: nat] :
                      ( ( Xb
                        = ( some_nat @ B ) )
                     => ( ( Y3
                          = ( some_nat @ ( X2 @ A @ B ) ) )
                       => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_8709_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8710_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8711_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_8712_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8713_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8714_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_8715_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_8716_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_8717_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_8718_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_8719_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8720_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_8721_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_8722_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8723_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_8724_sum__multicount__gen,axiom,
    ! [S2: set_o,T: set_o,R: $o > $o > $o,K: $o > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_o @ T )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8725_sum__multicount__gen,axiom,
    ! [S2: set_o,T: set_real,R: $o > real > $o,K: real > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_real @ T )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8726_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_o,R: real > $o > $o,K: $o > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( finite_finite_o @ T )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8727_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_real,R: real > real > $o,K: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( finite_finite_real @ T )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8728_sum__multicount__gen,axiom,
    ! [S2: set_o,T: set_int,R: $o > int > $o,K: int > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8729_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_int,R: real > int > $o,K: int > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8730_sum__multicount__gen,axiom,
    ! [S2: set_o,T: set_complex,R: $o > complex > $o,K: complex > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [J3: complex] :
                        ( ( member_complex @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups5693394587270226106ex_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8731_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_complex,R: real > complex > $o,K: complex > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [J3: complex] :
                        ( ( member_complex @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups5693394587270226106ex_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8732_sum__multicount__gen,axiom,
    ! [S2: set_o,T: set_Extended_enat,R: $o > extended_enat > $o,K: extended_enat > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite121521170596916366d_enat
                  @ ( collec4429806609662206161d_enat
                    @ ^ [J3: extended_enat] :
                        ( ( member_Extended_enat @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups2027974829824023292at_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8733_sum__multicount__gen,axiom,
    ! [S2: set_real,T: set_Extended_enat,R: real > extended_enat > $o,K: extended_enat > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S2 )
                        & ( R @ I4 @ X5 ) ) ) )
                = ( K @ X5 ) ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite121521170596916366d_enat
                  @ ( collec4429806609662206161d_enat
                    @ ^ [J3: extended_enat] :
                        ( ( member_Extended_enat @ J3 @ T )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S2 )
            = ( groups2027974829824023292at_nat @ K @ T ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_8734_sum__subtractf__nat,axiom,
    ! [A3: set_real,G3: real > nat,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ G3 @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8735_sum__subtractf__nat,axiom,
    ! [A3: set_o,G3: $o > nat,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups8507830703676809646_o_nat
          @ ^ [X: $o] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ G3 @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8736_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G3: set_nat > nat,F: set_nat > nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X: set_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( groups8294997508430121362at_nat @ G3 @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8737_sum__subtractf__nat,axiom,
    ! [A3: set_int,G3: int > nat,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8738_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G3: nat > nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
         => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G3 @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8739_sum__eq__Suc0__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A3 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: int] :
                  ( ( member_int @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8740_sum__eq__Suc0__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A3 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8741_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( groups977919841031483927at_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8742_sum__eq__Suc0__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: extended_enat] :
              ( ( member_Extended_enat @ X @ A3 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: extended_enat] :
                  ( ( member_Extended_enat @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8743_sum__eq__Suc0__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8744_sum__SucD,axiom,
    ! [F: nat > nat,A3: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A3 )
        = ( suc @ N ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A3 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_8745_sum__eq__1__iff,axiom,
    ! [A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X: int] :
              ( ( member_int @ X @ A3 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: int] :
                  ( ( member_int @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8746_sum__eq__1__iff,axiom,
    ! [A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A3 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8747_sum__eq__1__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( ( groups977919841031483927at_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8748_sum__eq__1__iff,axiom,
    ! [A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X: extended_enat] :
              ( ( member_Extended_enat @ X @ A3 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: extended_enat] :
                  ( ( member_Extended_enat @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8749_sum__eq__1__iff,axiom,
    ! [A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
          = one_one_nat )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A3 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8750_sum__multicount,axiom,
    ! [S: set_o,T2: set_o,R: $o > $o > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_o @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ T2 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_o @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8751_sum__multicount,axiom,
    ! [S: set_o,T2: set_real,R: $o > real > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_real @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ T2 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_real @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8752_sum__multicount,axiom,
    ! [S: set_real,T2: set_o,R: real > $o > $o,K: nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite_finite_o @ T2 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ T2 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_o
                  @ ( collect_o
                    @ ^ [J3: $o] :
                        ( ( member_o @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_o @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8753_sum__multicount,axiom,
    ! [S: set_real,T2: set_real,R: real > real > $o,K: nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite_finite_real @ T2 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ T2 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J3: real] :
                        ( ( member_real @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_real @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8754_sum__multicount,axiom,
    ! [S: set_o,T2: set_nat,R: $o > nat > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T2 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8755_sum__multicount,axiom,
    ! [S: set_real,T2: set_nat,R: real > nat > $o,K: nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T2 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J3: nat] :
                        ( ( member_nat @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8756_sum__multicount,axiom,
    ! [S: set_o,T2: set_int,R: $o > int > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T2 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8757_sum__multicount,axiom,
    ! [S: set_real,T2: set_int,R: real > int > $o,K: nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T2 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J3: int] :
                        ( ( member_int @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8758_sum__multicount,axiom,
    ! [S: set_o,T2: set_complex,R: $o > complex > $o,K: nat] :
      ( ( finite_finite_o @ S )
     => ( ( finite3207457112153483333omplex @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T2 )
             => ( ( finite_card_o
                  @ ( collect_o
                    @ ^ [I4: $o] :
                        ( ( member_o @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [I4: $o] :
                  ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [J3: complex] :
                        ( ( member_complex @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_complex @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8759_sum__multicount,axiom,
    ! [S: set_real,T2: set_complex,R: real > complex > $o,K: nat] :
      ( ( finite_finite_real @ S )
     => ( ( finite3207457112153483333omplex @ T2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T2 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I4: real] :
                        ( ( member_real @ I4 @ S )
                        & ( R @ I4 @ X5 ) ) ) )
                = K ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I4: real] :
                  ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [J3: complex] :
                        ( ( member_complex @ J3 @ T2 )
                        & ( R @ I4 @ J3 ) ) ) )
              @ S )
            = ( times_times_nat @ K @ ( finite_card_complex @ T2 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_8760_sum__diff__nat,axiom,
    ! [B2: set_complex,A3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8761_sum__diff__nat,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8762_sum__diff__nat,axiom,
    ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
       => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8763_sum__diff__nat,axiom,
    ! [B2: set_int,A3: set_int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A3 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8764_sum__diff__nat,axiom,
    ! [B2: set_nat,A3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A3 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8765_sum__diff1__nat,axiom,
    ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( ( member8440522571783428010at_nat @ A2 @ A3 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
          = ( groups977919841031483927at_nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8766_sum__diff1__nat,axiom,
    ! [A2: set_nat,A3: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat @ A2 @ A3 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member_set_nat @ A2 @ A3 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8767_sum__diff1__nat,axiom,
    ! [A2: real,A3: set_real,F: real > nat] :
      ( ( ( member_real @ A2 @ A3 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member_real @ A2 @ A3 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8768_sum__diff1__nat,axiom,
    ! [A2: $o,A3: set_o,F: $o > nat] :
      ( ( ( member_o @ A2 @ A3 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member_o @ A2 @ A3 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
          = ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8769_sum__diff1__nat,axiom,
    ! [A2: int,A3: set_int,F: int > nat] :
      ( ( ( member_int @ A2 @ A3 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member_int @ A2 @ A3 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8770_sum__diff1__nat,axiom,
    ! [A2: nat,A3: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A2 @ A3 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
      & ( ~ ( member_nat @ A2 @ A3 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8771_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8772_sum__Un__nat,axiom,
    ! [A3: set_int,B2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A3 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8773_sum__Un__nat,axiom,
    ! [A3: set_complex,B2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A3 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8774_sum__Un__nat,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( groups2027974829824023292at_nat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) @ ( groups2027974829824023292at_nat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8775_sum__Un__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( ( groups977919841031483927at_nat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) @ ( groups977919841031483927at_nat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8776_sum__Un__nat,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite4392333629123659920at_nat @ B2 )
       => ( ( groups1900718384385340925at_nat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups1900718384385340925at_nat @ F @ A3 ) @ ( groups1900718384385340925at_nat @ F @ B2 ) ) @ ( groups1900718384385340925at_nat @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8777_sum__Un__nat,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ A3 )
     => ( ( finite4343798906461161616at_nat @ B2 )
       => ( ( groups3860910324918113789at_nat @ F @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F @ A3 ) @ ( groups3860910324918113789at_nat @ F @ B2 ) ) @ ( groups3860910324918113789at_nat @ F @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8778_sum__Un__nat,axiom,
    ! [A3: set_nat,B2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8779_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8780_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8781_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8782_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_8783_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_8784_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A2: nat,B3: nat,Acc2: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B3 @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B3 @ A2 )
         => ( ( set_fo2584398358068434914at_nat @ F @ A2 @ B3 @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B3 @ A2 )
         => ( ( set_fo2584398358068434914at_nat @ F @ A2 @ B3 @ Acc2 )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B3 @ ( F @ A2 @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_8785_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
        = Y3 )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
               => ( Y3 = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa2 )
               => ( Y3
                  = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_8786_upto_Opinduct,axiom,
    ! [A0: int,A13: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A13 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A13 ) ) ) ).

% upto.pinduct
thf(fact_8787_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q6: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q6 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
                @ ( 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_8788_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q6: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q6 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
                @ ( 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_8789_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q6: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q6 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
                @ ( 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_8790_diffs__equiv,axiom,
    ! [C: nat > complex,X2: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
     => ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8791_diffs__equiv,axiom,
    ! [C: nat > real,X2: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8792_lessThan__eq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( set_ord_lessThan_nat @ X2 )
        = ( set_ord_lessThan_nat @ Y3 ) )
      = ( X2 = Y3 ) ) ).

% lessThan_eq_iff
thf(fact_8793_lessThan__eq__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ( set_ord_lessThan_int @ X2 )
        = ( set_ord_lessThan_int @ Y3 ) )
      = ( X2 = Y3 ) ) ).

% lessThan_eq_iff
thf(fact_8794_lessThan__eq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( set_or5984915006950818249n_real @ X2 )
        = ( set_or5984915006950818249n_real @ Y3 ) )
      = ( X2 = Y3 ) ) ).

% lessThan_eq_iff
thf(fact_8795_lessThan__iff,axiom,
    ! [I: $o,K: $o] :
      ( ( member_o @ I @ ( set_ord_lessThan_o @ K ) )
      = ( ord_less_o @ I @ K ) ) ).

% lessThan_iff
thf(fact_8796_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8797_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8798_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_8799_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8800_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_8801_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_8802_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_8803_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_8804_lessThan__subset__iff,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y3 ) )
      = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_8805_lessThan__subset__iff,axiom,
    ! [X2: num,Y3: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y3 ) )
      = ( ord_less_eq_num @ X2 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_8806_lessThan__subset__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
      = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_8807_lessThan__subset__iff,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y3 ) )
      = ( ord_less_eq_int @ X2 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_8808_lessThan__subset__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
      = ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_8809_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8810_sum_OlessThan__Suc,axiom,
    ! [G3: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8811_sum_OlessThan__Suc,axiom,
    ! [G3: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8812_sum_OlessThan__Suc,axiom,
    ! [G3: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8813_sum_OlessThan__Suc,axiom,
    ! [G3: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8814_single__Diff__lessThan,axiom,
    ! [K: $o] :
      ( ( minus_minus_set_o @ ( insert_o @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
      = ( insert_o @ K @ bot_bot_set_o ) ) ).

% single_Diff_lessThan
thf(fact_8815_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_8816_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_8817_single__Diff__lessThan,axiom,
    ! [K: real] :
      ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
      = ( insert_real @ K @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_8818_lessThan__non__empty,axiom,
    ! [X2: int] :
      ( ( set_ord_lessThan_int @ X2 )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_8819_lessThan__non__empty,axiom,
    ! [X2: real] :
      ( ( set_or5984915006950818249n_real @ X2 )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_8820_infinite__Iio,axiom,
    ! [A2: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A2 ) ) ).

% infinite_Iio
thf(fact_8821_infinite__Iio,axiom,
    ! [A2: real] :
      ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A2 ) ) ).

% infinite_Iio
thf(fact_8822_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_set_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8823_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8824_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8825_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8826_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8827_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8828_Iio__eq__empty__iff,axiom,
    ! [N: $o] :
      ( ( ( set_ord_lessThan_o @ N )
        = bot_bot_set_o )
      = ( N = bot_bot_o ) ) ).

% Iio_eq_empty_iff
thf(fact_8829_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_8830_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8831_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8832_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8833_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8834_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8835_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8836_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8837_finite__nat__bounded,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% finite_nat_bounded
thf(fact_8838_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8839_ivl__disj__int__one_I4_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(4)
thf(fact_8840_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_8841_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_8842_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_8843_sum_Onat__diff__reindex,axiom,
    ! [G3: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8844_sum_Onat__diff__reindex,axiom,
    ! [G3: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8845_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N: int] :
      ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_int @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8846_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N: real] :
      ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8847_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N: nat] :
      ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8848_Iio__Int__singleton,axiom,
    ! [X2: $o,K: $o] :
      ( ( ( ord_less_o @ X2 @ K )
       => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
          = ( insert_o @ X2 @ bot_bot_set_o ) ) )
      & ( ~ ( ord_less_o @ X2 @ K )
       => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
          = bot_bot_set_o ) ) ) ).

% Iio_Int_singleton
thf(fact_8849_Iio__Int__singleton,axiom,
    ! [X2: rat,K: rat] :
      ( ( ( ord_less_rat @ X2 @ K )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
          = ( insert_rat @ X2 @ bot_bot_set_rat ) ) )
      & ( ~ ( ord_less_rat @ X2 @ K )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
          = bot_bot_set_rat ) ) ) ).

% Iio_Int_singleton
thf(fact_8850_Iio__Int__singleton,axiom,
    ! [X2: num,K: num] :
      ( ( ( ord_less_num @ X2 @ K )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K ) @ ( insert_num @ X2 @ bot_bot_set_num ) )
          = ( insert_num @ X2 @ bot_bot_set_num ) ) )
      & ( ~ ( ord_less_num @ X2 @ K )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K ) @ ( insert_num @ X2 @ bot_bot_set_num ) )
          = bot_bot_set_num ) ) ) ).

% Iio_Int_singleton
thf(fact_8851_Iio__Int__singleton,axiom,
    ! [X2: nat,K: nat] :
      ( ( ( ord_less_nat @ X2 @ K )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
      & ( ~ ( ord_less_nat @ X2 @ K )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
          = bot_bot_set_nat ) ) ) ).

% Iio_Int_singleton
thf(fact_8852_Iio__Int__singleton,axiom,
    ! [X2: int,K: int] :
      ( ( ( ord_less_int @ X2 @ K )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
          = ( insert_int @ X2 @ bot_bot_set_int ) ) )
      & ( ~ ( ord_less_int @ X2 @ K )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
          = bot_bot_set_int ) ) ) ).

% Iio_Int_singleton
thf(fact_8853_Iio__Int__singleton,axiom,
    ! [X2: real,K: real] :
      ( ( ( ord_less_real @ X2 @ K )
       => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
          = ( insert_real @ X2 @ bot_bot_set_real ) ) )
      & ( ~ ( ord_less_real @ X2 @ K )
       => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
          = bot_bot_set_real ) ) ) ).

% Iio_Int_singleton
thf(fact_8854_suminf__le__const,axiom,
    ! [F: nat > int,X2: int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X2 ) ) ) ).

% suminf_le_const
thf(fact_8855_suminf__le__const,axiom,
    ! [F: nat > nat,X2: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X2 ) ) ) ).

% suminf_le_const
thf(fact_8856_suminf__le__const,axiom,
    ! [F: nat > real,X2: real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X2 ) ) ) ).

% suminf_le_const
thf(fact_8857_sum_OlessThan__Suc__shift,axiom,
    ! [G3: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G3 @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8858_sum_OlessThan__Suc__shift,axiom,
    ! [G3: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G3 @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8859_sum_OlessThan__Suc__shift,axiom,
    ! [G3: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G3 @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8860_sum_OlessThan__Suc__shift,axiom,
    ! [G3: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G3 @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8861_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8862_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8863_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8864_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8865_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8866_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8867_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X2: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8868_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8869_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X2: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8870_sum_OatLeast1__atMost__eq,axiom,
    ! [G3: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8871_sum_OatLeast1__atMost__eq,axiom,
    ! [G3: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8872_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F5: nat > nat > nat,A4: nat,B4: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F5 @ A4 @ Acc3 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_8873_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
        = Y3 )
     => ( ( ( ord_less_nat @ Xb @ Xa2 )
         => ( Y3 = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa2 )
         => ( Y3
            = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_8874_finite__enumerate__initial__segment,axiom,
    ! [S: set_Extended_enat,N: nat,S2: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ S @ ( set_or8419480210114673929d_enat @ S2 ) ) ) )
       => ( ( infini7641415182203889163d_enat @ ( inf_in8357106775501769908d_enat @ S @ ( set_or8419480210114673929d_enat @ S2 ) ) @ N )
          = ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ).

% finite_enumerate_initial_segment
thf(fact_8875_finite__enumerate__initial__segment,axiom,
    ! [S: set_nat,N: nat,S2: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) ) )
       => ( ( infini8530281810654367211te_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) @ N )
          = ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).

% finite_enumerate_initial_segment
thf(fact_8876_power__diff__1__eq,axiom,
    ! [X2: complex,N: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X2 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8877_power__diff__1__eq,axiom,
    ! [X2: rat,N: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8878_power__diff__1__eq,axiom,
    ! [X2: int,N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8879_power__diff__1__eq,axiom,
    ! [X2: real,N: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8880_one__diff__power__eq,axiom,
    ! [X2: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8881_one__diff__power__eq,axiom,
    ! [X2: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8882_one__diff__power__eq,axiom,
    ! [X2: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8883_one__diff__power__eq,axiom,
    ! [X2: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8884_geometric__sum,axiom,
    ! [X2: rat,N: nat] :
      ( ( X2 != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_8885_geometric__sum,axiom,
    ! [X2: complex,N: nat] :
      ( ( X2 != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_8886_geometric__sum,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_8887_sum__less__suminf,axiom,
    ! [F: nat > int,N: nat] :
      ( ( summable_int @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8888_sum__less__suminf,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( summable_nat @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8889_sum__less__suminf,axiom,
    ! [F: nat > real,N: nat] :
      ( ( summable_real @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8890_sum__gp__strict,axiom,
    ! [X2: rat,N: nat] :
      ( ( ( X2 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) )
      & ( ( X2 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8891_sum__gp__strict,axiom,
    ! [X2: complex,N: nat] :
      ( ( ( X2 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri8010041392384452111omplex @ N ) ) )
      & ( ( X2 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8892_sum__gp__strict,axiom,
    ! [X2: real,N: nat] :
      ( ( ( X2 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri5074537144036343181t_real @ N ) ) )
      & ( ( X2 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8893_diff__power__eq__sum,axiom,
    ! [X2: complex,N: nat,Y3: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X2 @ P5 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8894_diff__power__eq__sum,axiom,
    ! [X2: rat,N: nat,Y3: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X2 @ P5 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8895_diff__power__eq__sum,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X2 @ P5 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8896_diff__power__eq__sum,axiom,
    ! [X2: real,N: nat,Y3: real] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X2 @ P5 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8897_power__diff__sumr2,axiom,
    ! [X2: complex,N: nat,Y3: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X2 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8898_power__diff__sumr2,axiom,
    ! [X2: rat,N: nat,Y3: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X2 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8899_power__diff__sumr2,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X2 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8900_power__diff__sumr2,axiom,
    ! [X2: real,N: nat,Y3: real] :
      ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X2 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8901_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > rat,K4: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K4 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K4 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K4 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8902_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > int,K4: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K4 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8903_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > nat,K4: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K4 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8904_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > real,K4: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K4 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K4 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8905_sum__less__suminf2,axiom,
    ! [F: nat > int,N: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_8906_sum__less__suminf2,axiom,
    ! [F: nat > nat,N: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_8907_sum__less__suminf2,axiom,
    ! [F: nat > real,N: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_8908_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A2: nat,B3: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
        @ A2
        @ B3
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_8909_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A2: nat,B3: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
        @ A2
        @ B3
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_8910_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A2: nat,B3: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
        @ A2
        @ B3
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_8911_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A2: nat,B3: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
        @ A2
        @ B3
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_8912_one__diff__power__eq_H,axiom,
    ! [X2: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8913_one__diff__power__eq_H,axiom,
    ! [X2: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8914_one__diff__power__eq_H,axiom,
    ! [X2: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8915_one__diff__power__eq_H,axiom,
    ! [X2: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8916_termdiff__converges,axiom,
    ! [X2: real,K4: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K4 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K4 )
           => ( summable_real
              @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) ) )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8917_termdiff__converges,axiom,
    ! [X2: complex,K4: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K4 )
     => ( ! [X5: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K4 )
           => ( summable_complex
              @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) ) )
       => ( summable_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8918_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D6: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8919_in__measure,axiom,
    ! [X2: option_nat,Y3: option_nat,F: option_nat > nat] :
      ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X2 @ Y3 ) @ ( measure_option_nat @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_8920_in__measure,axiom,
    ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,F: set_Pr4329608150637261639at_nat > nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ ( measur4922264856574889999at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_8921_in__measure,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( measur1827424007717751593at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_8922_in__measure,axiom,
    ! [X2: nat,Y3: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_8923_in__measure,axiom,
    ! [X2: int,Y3: int,F: int > nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measure_int @ F ) )
      = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_8924_in__finite__psubset,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B2 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A3 @ B2 )
        & ( finite_finite_nat @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_8925_in__finite__psubset,axiom,
    ! [A3: set_int,B2: set_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A3 @ B2 ) @ finite_psubset_int )
      = ( ( ord_less_set_int @ A3 @ B2 )
        & ( finite_finite_int @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_8926_in__finite__psubset,axiom,
    ! [A3: set_complex,B2: set_complex] :
      ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A3 @ B2 ) @ finite8643634255014194347omplex )
      = ( ( ord_less_set_complex @ A3 @ B2 )
        & ( finite3207457112153483333omplex @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_8927_in__finite__psubset,axiom,
    ! [A3: set_Extended_enat,B2: set_Extended_enat] :
      ( ( member4453595087596390480d_enat @ ( produc6639060556116774935d_enat @ A3 @ B2 ) @ finite4251489430341359785d_enat )
      = ( ( ord_le2529575680413868914d_enat @ A3 @ B2 )
        & ( finite4001608067531595151d_enat @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_8928_in__finite__psubset,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A3 @ B2 ) @ finite4695646753290404266at_nat )
      = ( ( ord_le2604355607129572851at_nat @ A3 @ B2 )
        & ( finite4343798906461161616at_nat @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_8929_in__finite__psubset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B2 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A3 @ B2 )
        & ( finite6177210948735845034at_nat @ B2 ) ) ) ).

% in_finite_psubset
thf(fact_8930_monoI1,axiom,
    ! [X6: nat > real] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% monoI1
thf(fact_8931_monoI1,axiom,
    ! [X6: nat > set_int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_set_int @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X6 ) ) ).

% monoI1
thf(fact_8932_monoI1,axiom,
    ! [X6: nat > rat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_rat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% monoI1
thf(fact_8933_monoI1,axiom,
    ! [X6: nat > num] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_num @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% monoI1
thf(fact_8934_monoI1,axiom,
    ! [X6: nat > nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_nat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% monoI1
thf(fact_8935_monoI1,axiom,
    ! [X6: nat > int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_int @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% monoI1
thf(fact_8936_monoI2,axiom,
    ! [X6: nat > real] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% monoI2
thf(fact_8937_monoI2,axiom,
    ! [X6: nat > set_int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_set_int @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
     => ( topolo3100542954746470799et_int @ X6 ) ) ).

% monoI2
thf(fact_8938_monoI2,axiom,
    ! [X6: nat > rat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% monoI2
thf(fact_8939_monoI2,axiom,
    ! [X6: nat > num] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% monoI2
thf(fact_8940_monoI2,axiom,
    ! [X6: nat > nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% monoI2
thf(fact_8941_monoI2,axiom,
    ! [X6: nat > int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% monoI2
thf(fact_8942_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X8: nat > real] :
          ( ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
          | ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8943_monoseq__def,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X8: nat > set_int] :
          ( ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_set_int @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
          | ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8944_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X8: nat > rat] :
          ( ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
          | ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8945_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X8: nat > num] :
          ( ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_num @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
          | ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8946_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X8: nat > nat] :
          ( ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_nat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
          | ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8947_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X8: nat > int] :
          ( ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_int @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
          | ! [M2: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8948_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X8: nat > real] :
          ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8949_monoseq__Suc,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X8: nat > set_int] :
          ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8950_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X8: nat > rat] :
          ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8951_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X8: nat > num] :
          ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8952_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X8: nat > nat] :
          ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8953_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X8: nat > int] :
          ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8954_mono__SucI2,axiom,
    ! [X6: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% mono_SucI2
thf(fact_8955_mono__SucI2,axiom,
    ! [X6: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
     => ( topolo3100542954746470799et_int @ X6 ) ) ).

% mono_SucI2
thf(fact_8956_mono__SucI2,axiom,
    ! [X6: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% mono_SucI2
thf(fact_8957_mono__SucI2,axiom,
    ! [X6: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% mono_SucI2
thf(fact_8958_mono__SucI2,axiom,
    ! [X6: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% mono_SucI2
thf(fact_8959_mono__SucI2,axiom,
    ! [X6: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% mono_SucI2
thf(fact_8960_mono__SucI1,axiom,
    ! [X6: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% mono_SucI1
thf(fact_8961_mono__SucI1,axiom,
    ! [X6: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X6 ) ) ).

% mono_SucI1
thf(fact_8962_mono__SucI1,axiom,
    ! [X6: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% mono_SucI1
thf(fact_8963_mono__SucI1,axiom,
    ! [X6: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% mono_SucI1
thf(fact_8964_mono__SucI1,axiom,
    ! [X6: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% mono_SucI1
thf(fact_8965_mono__SucI1,axiom,
    ! [X6: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% mono_SucI1
thf(fact_8966_Maclaurin__exp__lt,axiom,
    ! [X2: real,N: nat] :
      ( ( X2 != 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 @ X2 ) )
            & ( ( exp_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8967_floor__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8968_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N2: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
          @ N2
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_8969_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
          @ N2
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_8970_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
          @ N2
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_8971_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N2: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
          @ N2
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_8972_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_8973_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8974_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_8975_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_8976_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_8977_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_8978_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_8979_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_8980_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_8981_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_8982_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8983_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8984_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_8985_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_8986_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_8987_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_8988_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_8989_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8990_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8991_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_8992_floor__eq,axiom,
    ! [N: int,X2: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = N ) ) ) ).

% floor_eq
thf(fact_8993_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_8994_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_8995_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_8996_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8997_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_8998_binomial__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).

% binomial_mono
thf(fact_8999_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9000_binomial__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K5 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9001_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_9002_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9003_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9004_floor__eq2,axiom,
    ! [N: int,X2: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = N ) ) ) ).

% floor_eq2
thf(fact_9005_floor__divide__real__eq__div,axiom,
    ! [B3: int,A2: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B3 )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A2 @ ( ring_1_of_int_real @ B3 ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A2 ) @ B3 ) ) ) ).

% floor_divide_real_eq_div
thf(fact_9006_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9007_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K5 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9008_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9009_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9010_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_9011_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_9012_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B3: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B3 @ X2 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B3 @ ( ring_1_of_int_real @ K ) ) @ X2 )
            & ( ord_less_real @ X2 @ ( powr_real @ B3 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9013_Maclaurin__exp__le,axiom,
    ! [X2: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
      & ( ( exp_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9014_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_9015_floor__log__nat__eq__if,axiom,
    ! [B3: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_9016_powr__int,axiom,
    ! [X2: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X2 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9017_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_9018_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_9019_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_9020_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_9021_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_9022_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9023_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9024_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9025_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9026_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W2 @ Z ) ) ) ).

% zless_nat_conj
thf(fact_9027_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9028_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9029_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_9030_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9031_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_9032_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_9033_nat__ceiling__le__eq,axiom,
    ! [X2: real,A2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A2 )
      = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9034_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_9035_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_9036_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_9037_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9038_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9039_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X2: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9040_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: num,N: nat,A2: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9041_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_9042_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
        & ( zero_zero_nat != A2 ) ) ).

% gcd_nat.extremum_strict
thf(fact_9043_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
      = ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_9044_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
        & ( A2 != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_9045_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
     => ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_9046_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_9047_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_9048_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_9049_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9050_nat__mono,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ X2 @ Y3 )
     => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ).

% nat_mono
thf(fact_9051_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9052_eq__nat__nat__iff,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z7 ) )
          = ( Z = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_9053_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X3: nat] : ( P2 @ X3 ) )
    = ( ^ [P3: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_9054_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X3: nat] : ( P2 @ X3 ) )
    = ( ^ [P3: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_9055_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9056_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_9057_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_9058_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_9059_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9060_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_9061_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_9062_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_9063_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_9064_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_9065_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_9066_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_9067_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_mono_iff
thf(fact_9068_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_9069_nat__le__iff,axiom,
    ! [X2: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
      = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_9070_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_9071_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_9072_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_9073_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_9074_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_9075_bezout__add__strong__nat,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D6: nat,X5: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D6 @ A2 )
          & ( dvd_dvd_nat @ D6 @ B3 )
          & ( ( times_times_nat @ A2 @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ D6 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_9076_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_9077_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_9078_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_9079_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q3 )
          = ( modulo_modulo_nat @ N @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_9080_real__nat__ceiling__ge,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9081_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_9082_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W2 @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_9083_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W2 @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_9084_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_9085_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ( nat2 @ W2 )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_9086_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N2: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ( P @ N2 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9087_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_9088_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9089_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9090_nat__diff__distrib_H,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( minus_minus_int @ X2 @ Y3 ) )
          = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9091_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9092_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9093_nat__div__distrib,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib
thf(fact_9094_nat__div__distrib_H,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib'
thf(fact_9095_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_9096_nat__floor__neg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9097_nat__mod__distrib,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y3 ) )
          = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9098_floor__eq3,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_9099_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_9100_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_9101_le__nat__floor,axiom,
    ! [X2: nat,A2: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A2 )
     => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) ) ) ).

% le_nat_floor
thf(fact_9102_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_9103_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_9104_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_9105_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_9106_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9107_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_9108_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_9109_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9110_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_9111_nat__abs__int__diff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( ord_less_eq_nat @ A2 @ B3 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
          = ( minus_minus_nat @ B3 @ A2 ) ) )
      & ( ~ ( ord_less_eq_nat @ A2 @ B3 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
          = ( minus_minus_nat @ A2 @ B3 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9112_floor__eq4,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_9113_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_9114_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9115_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9116_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).

% even_set_encode_iff
thf(fact_9117_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_9118_polynomial__product__nat,axiom,
    ! [M: nat,A2: nat > nat,N: nat,B3: nat > nat,X2: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A2 @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B3 @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( power_power_nat @ X2 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B3 @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A2 @ K3 ) @ ( B3 @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X2 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9119_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_9120_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q6: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9121_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q6: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9122_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_9123_Bernoulli__inequality__even,axiom,
    ! [N: nat,X2: 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 ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_9124_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_9125_vebt__buildup_Oelims,axiom,
    ! [X2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y3
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_9126_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M2: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M2 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q6: nat] : ( product_Pair_nat_nat @ ( suc @ Q6 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9127_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_9128_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_9129_sin__x__le__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).

% sin_x_le_x
thf(fact_9130_sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).

% sin_le_one
thf(fact_9131_cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).

% cos_le_one
thf(fact_9132_abs__sin__x__le__abs__x,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_9133_sin__cos__le1,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_9134_sin__x__ge__neg__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_9135_sin__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).

% sin_ge_minus_one
thf(fact_9136_cos__ge__minus__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).

% cos_ge_minus_one
thf(fact_9137_abs__sin__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_9138_abs__cos__le__one,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_9139_sin__gt__zero__02,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_9140_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_9141_cos__is__zero,axiom,
    ? [X5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X5 )
      & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X5 )
        = zero_zero_real )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X5 ) ) ) ).

% cos_is_zero
thf(fact_9142_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_9143_cos__double__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_9144_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_9145_vebt__buildup_Opelims,axiom,
    ! [X2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y3 )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y3
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y3
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y3
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_9146_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X2 )
            & ( ( sin_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
                  @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_9147_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X2 )
          & ( ( sin_real @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
                @ ( 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 @ X2 @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_9148_Maclaurin__sin__expansion2,axiom,
    ! [X2: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
      & ( ( sin_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
            @ ( 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 @ X2 @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9149_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_9150_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_9151_sin__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero
thf(fact_9152_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_9153_cos__inj__pi,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ( cos_real @ X2 )
                = ( cos_real @ Y3 ) )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_9154_cos__mono__le__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_9155_cos__monotone__0__pi__le,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_9156_sin__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_ge_zero
thf(fact_9157_cos__monotone__0__pi,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ Y3 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_9158_cos__mono__less__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_real @ Y3 @ X2 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_9159_sin__eq__0__pi,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
     => ( ( ord_less_real @ X2 @ pi )
       => ( ( ( sin_real @ X2 )
            = zero_zero_real )
         => ( X2 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_9160_cos__monotone__minus__pi__0_H,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_9161_sincos__principal__value,axiom,
    ! [X2: real] :
    ? [Y4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
      & ( ord_less_eq_real @ Y4 @ pi )
      & ( ( sin_real @ Y4 )
        = ( sin_real @ X2 ) )
      & ( ( cos_real @ Y4 )
        = ( cos_real @ X2 ) ) ) ).

% sincos_principal_value
thf(fact_9162_sin__zero__pi__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ pi )
     => ( ( ( sin_real @ X2 )
          = zero_zero_real )
        = ( X2 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_9163_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_9164_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_9165_cos__monotone__minus__pi__0,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_9166_cos__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
            & ( ord_less_eq_real @ X5 @ pi )
            & ( ( cos_real @ X5 )
              = Y3 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y3 ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% cos_total
thf(fact_9167_pi__half__less__two,axiom,
    ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_less_two
thf(fact_9168_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_9169_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_9170_pi__half__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_gt_zero
thf(fact_9171_cos__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_gt_zero
thf(fact_9172_sin__gt__zero2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero2
thf(fact_9173_sin__lt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ pi @ X2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_9174_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_9175_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_9176_sin__inj__pi,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X2 )
                = ( sin_real @ Y3 ) )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_9177_sin__mono__le__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_9178_sin__monotone__2pi__le,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_9179_arctan__ubound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_9180_sin__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ pi @ X2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_9181_minus__pi__half__less__zero,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).

% minus_pi_half_less_zero
thf(fact_9182_cos__gt__zero__pi,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_9183_sin__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_9184_cos__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_ge_zero
thf(fact_9185_sin__mono__less__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_9186_sin__monotone__2pi,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_9187_sin__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
            & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X5 )
              = Y3 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y5 )
                    = Y3 ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% sin_total
thf(fact_9188_arctan__lbound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).

% arctan_lbound
thf(fact_9189_arctan__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_9190_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_9191_sincos__total__pi,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ pi )
            & ( X2
              = ( cos_real @ T6 ) )
            & ( Y3
              = ( sin_real @ T6 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_9192_sincos__total__pi__half,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T6: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X2
                = ( cos_real @ T6 ) )
              & ( Y3
                = ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_9193_sincos__total__2pi__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) )
          & ( X2
            = ( cos_real @ T6 ) )
          & ( Y3
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_9194_cos__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( cos_real @ X2 )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_9195_sin__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( sin_real @ X2 )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_9196_sincos__total__2pi,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( 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 ) )
             => ( ( X2
                  = ( cos_real @ T6 ) )
               => ( Y3
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_9197_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X2 )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
                  @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_9198_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X2 @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
                  @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_9199_Maclaurin__cos__expansion,axiom,
    ! [X2: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
      & ( ( cos_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
            @ ( 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 @ X2 @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9200_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_9201_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_9202_lemma__tan__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X5: real] :
          ( ( ord_less_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).

% lemma_tan_total
thf(fact_9203_tan__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).

% tan_gt_zero
thf(fact_9204_lemma__tan__total1,axiom,
    ! [Y3: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y3 ) ) ).

% lemma_tan_total1
thf(fact_9205_tan__mono__lt__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_9206_tan__monotone_H,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
         => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y3 @ X2 )
              = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_9207_tan__monotone,axiom,
    ! [Y3: real,X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X2 )
       => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ).

% tan_monotone
thf(fact_9208_tan__total,axiom,
    ! [Y3: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y3 )
      & ! [Y5: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
            & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y5 )
              = Y3 ) )
         => ( Y5 = X5 ) ) ) ).

% tan_total
thf(fact_9209_tan__pos__pi2__le,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_9210_tan__total__pos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X5 )
            = Y3 ) ) ) ).

% tan_total_pos
thf(fact_9211_tan__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_9212_tan__mono__le__eq,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_9213_tan__mono__le,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ).

% tan_mono_le
thf(fact_9214_tan__bound__pi2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_9215_arctan,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y3 ) )
        = Y3 ) ) ).

% arctan
thf(fact_9216_arctan__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X2 ) )
          = X2 ) ) ) ).

% arctan_tan
thf(fact_9217_arctan__unique,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X2 )
            = Y3 )
         => ( ( arctan @ Y3 )
            = X2 ) ) ) ) ).

% arctan_unique
thf(fact_9218_tan__total__pi4,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ? [Z4: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
          & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z4 )
            = X2 ) ) ) ).

% tan_total_pi4
thf(fact_9219_sin__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X2 )
        = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_9220_cos__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X2 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_9221_Maclaurin__sin__bound,axiom,
    ! [X2: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X2 )
          @ ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9222_cot__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9223_real__sqrt__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ X2 @ Y3 ) ) ).

% real_sqrt_less_iff
thf(fact_9224_real__sqrt__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ X2 @ Y3 ) ) ).

% real_sqrt_le_iff
thf(fact_9225_real__sqrt__gt__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_9226_real__sqrt__lt__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_9227_real__sqrt__ge__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_ge_0_iff
thf(fact_9228_real__sqrt__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_9229_real__sqrt__lt__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
      = ( ord_less_real @ X2 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_9230_real__sqrt__gt__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_9231_real__sqrt__le__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
      = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_9232_real__sqrt__ge__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_ge_1_iff
thf(fact_9233_real__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 ) ) ).

% real_sqrt_pow2
thf(fact_9234_real__sqrt__pow2__iff,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% real_sqrt_pow2_iff
thf(fact_9235_real__sqrt__le__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ Y3 )
     => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_le_mono
thf(fact_9236_real__sqrt__less__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_less_mono
thf(fact_9237_sqrt__divide__self__eq,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
        = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9238_real__sqrt__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_9239_real__sqrt__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_9240_real__sqrt__eq__zero__cancel,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( sqrt @ X2 )
          = zero_zero_real )
       => ( X2 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_9241_real__sqrt__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_one
thf(fact_9242_real__inv__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X2 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9243_real__div__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
        = ( sqrt @ X2 ) ) ) ).

% real_div_sqrt
thf(fact_9244_sqrt__add__le__add__sqrt,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_9245_le__real__sqrt__sumsq,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_9246_inverse__powr,axiom,
    ! [Y3: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A2 )
        = ( inverse_inverse_real @ ( powr_real @ Y3 @ A2 ) ) ) ) ).

% inverse_powr
thf(fact_9247_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_9248_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D6: real,E: real] :
          ( ( ord_less_real @ D6 @ E )
         => ( ( P @ D6 )
           => ( P @ E ) ) )
     => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9249_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D6: real,E: real] :
          ( ( ord_less_real @ D6 @ E )
         => ( ( P @ D6 )
           => ( P @ E ) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_9250_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N2: nat] :
            ( ( N2 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_9251_ln__inverse,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X2 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_inverse
thf(fact_9252_real__less__rsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_real @ X2 @ ( sqrt @ Y3 ) ) ) ).

% real_less_rsqrt
thf(fact_9253_real__le__rsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_eq_real @ X2 @ ( sqrt @ Y3 ) ) ) ).

% real_le_rsqrt
thf(fact_9254_sqrt__le__D,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 )
     => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_9255_log__inverse,axiom,
    ! [A2: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( log @ A2 @ ( inverse_inverse_real @ X2 ) )
            = ( uminus_uminus_real @ ( log @ A2 @ X2 ) ) ) ) ) ) ).

% log_inverse
thf(fact_9256_real__le__lsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).

% real_le_lsqrt
thf(fact_9257_real__sqrt__unique,axiom,
    ! [Y3: real,X2: real] :
      ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( sqrt @ X2 )
          = Y3 ) ) ) ).

% real_sqrt_unique
thf(fact_9258_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_9259_real__sqrt__sum__squares__ge1,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_9260_real__sqrt__sum__squares__ge2,axiom,
    ! [Y3: real,X2: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_9261_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A2 @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B3 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_9262_sqrt__ge__absD,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y3 ) )
     => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).

% sqrt_ge_absD
thf(fact_9263_exp__plus__inverse__exp,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9264_real__less__lsqrt,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).

% real_less_lsqrt
thf(fact_9265_sqrt__sum__squares__le__sum,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_9266_real__sqrt__ge__abs1,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_9267_real__sqrt__ge__abs2,axiom,
    ! [Y3: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_9268_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_9269_ln__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( sqrt @ X2 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_9270_plus__inverse__ge__2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9271_arsinh__real__aux,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_9272_real__sqrt__power__even,axiom,
    ! [N: nat,X2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( sqrt @ X2 ) @ N )
          = ( power_power_real @ X2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_9273_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X2: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_9274_arith__geo__mean__sqrt,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_9275_powr__half__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X2 ) ) ) ).

% powr_half_sqrt
thf(fact_9276_real__le__x__sinh,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9277_real__le__abs__sinh,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9278_cos__x__y__le__one,axiom,
    ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_9279_real__sqrt__sum__squares__less,axiom,
    ! [X2: real,U: real,Y3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_9280_arcosh__real__def,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( arcosh_real @ X2 )
        = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_9281_powr__real__of__int,axiom,
    ! [X2: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X2 @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9282_sinh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9283_cot__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).

% cot_gt_zero
thf(fact_9284_sqrt__sum__squares__half__less,axiom,
    ! [X2: real,U: real,Y3: real] :
      ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_9285_sin__cos__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
     => ( ( sin_real @ X2 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_9286_cos__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X2 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_9287_sin__arccos__abs,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y3 ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_9288_sin__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X2 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_9289_le__arcsin__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X2 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X2 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_9290_cos__arccos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y3 ) )
          = Y3 ) ) ) ).

% cos_arccos
thf(fact_9291_sin__arcsin,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y3 ) )
          = Y3 ) ) ) ).

% sin_arcsin
thf(fact_9292_cosh__real__pos,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_pos
thf(fact_9293_arcosh__cosh__real,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( arcosh_real @ ( cosh_real @ X2 ) )
        = X2 ) ) ).

% arcosh_cosh_real
thf(fact_9294_cosh__real__nonpos__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9295_cosh__real__nonneg__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9296_cosh__real__nonneg,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_nonneg
thf(fact_9297_cosh__real__ge__1,axiom,
    ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_ge_1
thf(fact_9298_sinh__less__cosh__real,axiom,
    ! [X2: real] : ( ord_less_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).

% sinh_less_cosh_real
thf(fact_9299_sinh__le__cosh__real,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).

% sinh_le_cosh_real
thf(fact_9300_cosh__real__strict__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ Y3 )
       => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9301_cosh__real__nonneg__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9302_cosh__real__nonpos__less__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9303_arccos__le__arccos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_9304_arccos__eq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
     => ( ( ( arccos @ X2 )
          = ( arccos @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% arccos_eq_iff
thf(fact_9305_arccos__le__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
          = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).

% arccos_le_mono
thf(fact_9306_arcsin__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
          = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).

% arcsin_minus
thf(fact_9307_arcsin__le__arcsin,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_9308_arcsin__eq__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ( arcsin @ X2 )
            = ( arcsin @ Y3 ) )
          = ( X2 = Y3 ) ) ) ) ).

% arcsin_eq_iff
thf(fact_9309_arcsin__le__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).

% arcsin_le_mono
thf(fact_9310_arccos__lbound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).

% arccos_lbound
thf(fact_9311_arccos__less__arccos,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_9312_arccos__less__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
          = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).

% arccos_less_mono
thf(fact_9313_arccos__ubound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_9314_arccos__cos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( arccos @ ( cos_real @ X2 ) )
          = X2 ) ) ) ).

% arccos_cos
thf(fact_9315_arcsin__less__arcsin,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_9316_arcsin__less__mono,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).

% arcsin_less_mono
thf(fact_9317_cos__arccos__abs,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y3 ) )
        = Y3 ) ) ).

% cos_arccos_abs
thf(fact_9318_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_9319_arccos__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_9320_arccos__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_9321_sin__arccos__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X2 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_9322_arccos__cos2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
       => ( ( arccos @ ( cos_real @ X2 ) )
          = ( uminus_uminus_real @ X2 ) ) ) ) ).

% arccos_cos2
thf(fact_9323_arccos__minus,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_minus
thf(fact_9324_cos__arcsin__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X2 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_9325_arccos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
          & ( ( cos_real @ ( arccos @ Y3 ) )
            = Y3 ) ) ) ) ).

% arccos
thf(fact_9326_arccos__minus__abs,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).

% arccos_minus_abs
thf(fact_9327_arccos__le__pi2,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_9328_cosh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9329_arcsin__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_9330_arcsin__lbound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).

% arcsin_lbound
thf(fact_9331_arcsin__ubound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_9332_arcsin__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_9333_arcsin__sin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X2 ) )
          = X2 ) ) ) ).

% arcsin_sin
thf(fact_9334_arcsin,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y3 ) )
            = Y3 ) ) ) ) ).

% arcsin
thf(fact_9335_arcsin__pi,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y3 ) )
            = Y3 ) ) ) ) ).

% arcsin_pi
thf(fact_9336_arcsin__le__iff,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y3 )
              = ( ord_less_eq_real @ X2 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_9337_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq_int @ D5 @ Z2 )
                & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9338_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z2: int] :
                ( ( ord_less_eq_int @ D5 @ Z8 )
                & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9339_set__decode__0,axiom,
    ! [X2: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).

% set_decode_0
thf(fact_9340_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_9341_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_9342_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
        = A3 ) ) ).

% set_encode_inverse
thf(fact_9343_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_9344_ln__neg__is__const,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ln_ln_real @ X2 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9345_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% subset_decode_imp_le
thf(fact_9346_arccos__def,axiom,
    ( arccos
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y ) ) ) ) ) ).

% arccos_def
thf(fact_9347_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9348_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9349_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X )
                = Y ) ) ) ) ) ).

% arcsin_def
thf(fact_9350_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9351_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9352_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_9353_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_9354_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_9355_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9356_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_9357_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9358_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9359_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9360_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_9361_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9362_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9363_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9364_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9365_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_9366_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_9367_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_9368_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9369_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9370_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9371_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_9372_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9373_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_9374_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9375_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9376_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9377_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9378_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9379_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_9380_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_9381_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9382_and__int_Opelims,axiom,
    ! [X2: int,Xa2: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y3
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y3
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( 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 @ X2 @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_9383_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9384_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_9385_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9386_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9387_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_rat_def
thf(fact_9388_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9389_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_rat_def
thf(fact_9390_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R2
                 != ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9391_AND__lower,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) ) ) ).

% AND_lower
thf(fact_9392_AND__upper1,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ X2 ) ) ).

% AND_upper1
thf(fact_9393_AND__upper2,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Y3 ) ) ).

% AND_upper2
thf(fact_9394_AND__upper1_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9395_AND__upper2_H,axiom,
    ! [Y3: int,Z: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9396_AND__upper2_H_H,axiom,
    ! [Y3: int,Z: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9397_AND__upper1_H_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9398_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_9399_and__int_Oelims,axiom,
    ! [X2: int,Xa2: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
        = Y3 )
     => ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y3
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y3
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9400_normalize__negative,axiom,
    ! [Q3: int,P6: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_9401_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_9402_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_9403_and__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_9404_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_9405_and__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_9406_and__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_9407_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_9408_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_9409_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9410_xor__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9411_xor__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9412_xor__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9413_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P6: int,Q3: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P6 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_9414_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N2: nat] :
          ( if_nat
          @ ( ( M2 = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9415_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9416_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_9417_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_9418_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9419_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9420_XOR__lower,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) ) ) ) ).

% XOR_lower
thf(fact_9421_Least__Suc2,axiom,
    ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
      ( ( P @ N )
     => ( ( Q @ M )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P @ ( suc @ K2 ) )
                = ( Q @ K2 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_9422_Least__Suc,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_9423_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9424_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9425_XOR__upper,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9426_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9427_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9428_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_9429_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_9430_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9431_or__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(4)
thf(fact_9432_or__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(2)
thf(fact_9433_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(1)
thf(fact_9434_or__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(3)
thf(fact_9435_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
           => ( P @ M2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9436_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
            & ( P @ M2 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9437_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9438_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9439_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9440_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_9441_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9442_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_9443_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_9444_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_9445_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
     => ( A3
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9446_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9447_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9448_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9449_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_9450_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S ) ) ).

% card_sum_le_nat_sum
thf(fact_9451_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_9452_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_9453_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_9454_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9455_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: nat > nat,B3: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( B3 @ I4 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9456_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M2: nat] :
          ( ( ord_less_eq_nat @ M8 @ M2 )
         => ! [N2: nat] :
              ( ( ord_less_eq_nat @ M8 @ N2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9457_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_9458_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9459_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9460_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_9461_OR__lower,axiom,
    ! [X2: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) ) ) ) ).

% OR_lower
thf(fact_9462_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_9463_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9464_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_9465_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_9466_OR__upper,axiom,
    ! [X2: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_9467_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_9468_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_9469_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9470_upto_Opelims,axiom,
    ! [X2: int,Xa2: int,Y3: list_int] :
      ( ( ( upto @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa2 )
               => ( Y3
                  = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
               => ( Y3 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9471_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9472_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9473_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9474_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9475_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9476_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9477_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9478_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9479_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9480_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9481_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9482_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9483_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9484_upto_Oelims,axiom,
    ! [X2: int,Xa2: int,Y3: list_int] :
      ( ( ( upto @ X2 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_int @ X2 @ Xa2 )
         => ( Y3
            = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
         => ( Y3 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9485_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9486_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q6: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9487_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9488_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9489_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9490_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_9491_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9492_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_9493_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_9494_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X2: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
      = ( ord_less_int @ Xa2 @ X2 ) ) ).

% less_integer.abs_eq
thf(fact_9495_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9496_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_9497_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9498_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_9499_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X2: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
      = ( ord_less_eq_int @ Xa2 @ X2 ) ) ).

% less_eq_integer.abs_eq
thf(fact_9500_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_9501_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9502_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N3 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K @ M3 )
                = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9503_binomial__def,axiom,
    ( binomial
    = ( ^ [N2: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K6: set_nat] :
                ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
                & ( ( finite_card_nat @ K6 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9504_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9505_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
          & ! [N5: nat] :
              ( ( ord_less_nat @ N5 @ ( finite_card_nat @ S ) )
             => ( member_nat @ ( R3 @ N5 ) @ S ) ) ) ) ).

% finite_enumerate
thf(fact_9506_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9507_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9508_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).

% zero_notin_Suc_image
thf(fact_9509_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9510_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_iff
thf(fact_9511_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_9512_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_9513_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9514_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9515_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B3: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).

% image_int_atLeastAtMost
thf(fact_9516_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B3: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B3 ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9517_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_9518_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_9519_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_9520_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_9521_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_9522_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_9523_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X: int] : ( plus_plus_int @ X @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9524_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_9525_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y3: nat,X2: nat] :
      ( ( ( ord_less_nat @ C @ Y3 )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y3 )
       => ( ( ( ord_less_nat @ X2 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X2 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9526_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9527_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9528_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9529_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_9530_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_9531_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).

% Inf_nat_def1
thf(fact_9532_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_9533_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_9534_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_9535_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_9536_range__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat )
        = S ) ) ).

% range_enumerate
thf(fact_9537_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_9538_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9539_suminf__eq__SUP__real,axiom,
    ! [X6: nat > real] :
      ( ( summable_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X6 @ I2 ) )
       => ( ( suminf_real @ X6 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X6 @ ( set_ord_lessThan_nat @ I4 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9540_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9541_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A2 != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9542_root__def,axiom,
    ( root
    = ( ^ [N2: nat,X: real] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9543_DERIV__even__real__root,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X2 @ 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 @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9544_DERIV__real__root__generic,axiom,
    ! [N: nat,X2: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X2 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X2 )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X2 @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9545_DERIV__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ 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 @ X2 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9546_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D6 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9547_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D6 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9548_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D6 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9549_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X2: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
                 => ( ( ord_less_real @ H4 @ D6 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9550_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B3: real,G3: real > real,G4: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
         => ( has_fi5821293074295781190e_real @ G3 @ ( G4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G4 @ X5 ) ) )
       => ( ( ord_less_eq_real @ A2 @ B3 )
         => ( ord_less_eq_real @ ( G3 @ A2 ) @ ( G3 @ B3 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9551_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A2 @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B3 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9552_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A2 @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B3 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9553_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A2 @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B3 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9554_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A2 @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B3 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9555_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D6 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9556_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D6 )
                 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9557_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D6 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9558_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D6 )
                 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9559_MVT2,axiom,
    ! [A2: real,B3: real,F: real > real,F6: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A2 @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B3 )
             => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
       => ? [Z4: real] :
            ( ( ord_less_real @ A2 @ Z4 )
            & ( ord_less_real @ Z4 @ B3 )
            & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
              = ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ ( F6 @ Z4 ) ) ) ) ) ) ).

% MVT2
thf(fact_9560_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
             => ( ( F @ X2 )
                = ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9561_DERIV__ln,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9562_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9563_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9564_DERIV__ln__divide,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9565_DERIV__pow,axiom,
    ! [N: nat,X2: real,S2: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ).

% DERIV_pow
thf(fact_9566_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z2: real] : ( powr_real @ Z2 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9567_DERIV__log,axiom,
    ! [X2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ( log @ B3 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B3 ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9568_DERIV__fun__powr,axiom,
    ! [G3: real > real,M: real,X2: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G3 @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G3 @ X2 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G3 @ X ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G3 @ X2 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9569_DERIV__powr,axiom,
    ! [G3: real > real,M: real,X2: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G3 @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G3 @ X2 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G3 @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G3 @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G3 @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G3 @ X2 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9570_DERIV__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9571_DERIV__real__sqrt__generic,axiom,
    ! [X2: real,D4: real] :
      ( ( X2 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X2 @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9572_arcosh__real__has__field__derivative,axiom,
    ! [X2: real,A3: set_real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A3 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9573_artanh__real__has__field__derivative,axiom,
    ! [X2: real,A3: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A3 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9574_DERIV__real__root,axiom,
    ! [N: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9575_DERIV__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9576_DERIV__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9577_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
            & ( ( F @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9578_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
          & ( ( F @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9579_DERIV__odd__real__root,axiom,
    ! [N: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X2 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9580_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9581_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9582_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H2 @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9583_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X2 != zero_zero_real )
         => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ 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 @ X2 ) )
                & ( ( F @ X2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9584_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T6: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
            & ( ( F @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9585_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A2 @ T6 )
                & ( ord_less_eq_real @ T6 @ B3 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A2 @ C )
           => ( ( ord_less_eq_real @ C @ B3 )
             => ( ( ord_less_eq_real @ A2 @ X2 )
               => ( ( ord_less_eq_real @ X2 @ B3 )
                 => ( ( X2 != C )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X2 @ C )
                         => ( ( ord_less_real @ X2 @ T6 )
                            & ( ord_less_real @ T6 @ C ) ) )
                        & ( ~ ( ord_less_real @ X2 @ C )
                         => ( ( ord_less_real @ C @ T6 )
                            & ( ord_less_real @ T6 @ X2 ) ) )
                        & ( ( F @ X2 )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ M2 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9586_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A2 @ T6 )
                & ( ord_less_eq_real @ T6 @ B3 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A2 @ C )
           => ( ( ord_less_real @ C @ B3 )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B3 )
                  & ( ( F @ B3 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ M2 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9587_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A2 @ T6 )
                & ( ord_less_eq_real @ T6 @ B3 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A2 @ C )
           => ( ( ord_less_eq_real @ C @ B3 )
             => ? [T6: real] :
                  ( ( ord_less_real @ A2 @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ M2 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9588_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
      ( ! [M4: nat,T6: real] :
          ( ( ( ord_less_nat @ M4 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M3: nat,T7: real] :
            ( ( ( ord_less_nat @ M3 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( 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 @ B2 @ ( 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 @ B2 @ ( 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_9589_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X5 @ N2 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] :
                ( suminf_real
                @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9590_DERIV__isconst3,axiom,
    ! [A2: real,B3: real,X2: real,Y3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
       => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
           => ( ( F @ X2 )
              = ( F @ Y3 ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9591_DERIV__series_H,axiom,
    ! [F: real > nat > real,F6: real > nat > real,X0: real,A2: real,B3: real,L5: nat > real] :
      ( ! [N3: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( F @ X @ N3 )
          @ ( F6 @ X0 @ N3 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
           => ( summable_real @ ( F @ X5 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
         => ( ( summable_real @ ( F6 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N3: nat,X5: real,Y4: real] :
                    ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
                   => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y4 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
                  @ ( suminf_real @ ( F6 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9592_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
                  @ ^ [M2: nat] :
                      ( collect_nat
                      @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) )
                  @ M5 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9593_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9594_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9595_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_9596_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_9597_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_9598_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9599_isCont__Lb__Ub,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A2 @ X5 )
              & ( ord_less_eq_real @ X5 @ B3 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: real] :
            ( ! [X4: real] :
                ( ( ( ord_less_eq_real @ A2 @ X4 )
                  & ( ord_less_eq_real @ X4 @ B3 ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X4 ) )
                  & ( ord_less_eq_real @ ( F @ X4 ) @ M9 ) ) )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ M9 ) )
               => ? [X5: real] :
                    ( ( ord_less_eq_real @ A2 @ X5 )
                    & ( ord_less_eq_real @ X5 @ B3 )
                    & ( ( F @ X5 )
                      = Y5 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9600_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9601_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ( F @ X4 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9602_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9603_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_9604_isCont__inverse__function2,axiom,
    ! [A2: real,X2: real,B3: real,G3: real > real,F: real > real] :
      ( ( ord_less_real @ A2 @ X2 )
     => ( ( ord_less_real @ X2 @ B3 )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A2 @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B3 )
               => ( ( G3 @ ( F @ Z4 ) )
                  = Z4 ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_eq_real @ A2 @ Z4 )
               => ( ( ord_less_eq_real @ Z4 @ B3 )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G3 ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9605_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).

% Sup_nat_def
thf(fact_9606_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9607_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9608_isCont__arcosh,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9609_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G3: real > real,X2: real,A2: real,B3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G3 @ X2 ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A2 @ X2 )
         => ( ( ord_less_real @ X2 @ B3 )
           => ( ! [Y4: real] :
                  ( ( ord_less_real @ A2 @ Y4 )
                 => ( ( ord_less_real @ Y4 @ B3 )
                   => ( ( F @ ( G3 @ Y4 ) )
                      = Y4 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G3 )
               => ( has_fi5821293074295781190e_real @ G3 @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9610_isCont__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9611_isCont__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9612_LIM__less__bound,axiom,
    ! [B3: real,X2: real,F: real > real] :
      ( ( ord_less_real @ B3 @ X2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B3 @ X2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9613_isCont__artanh,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9614_isCont__inverse__function,axiom,
    ! [D: real,X2: real,G3: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
           => ( ( G3 @ ( F @ Z4 ) )
              = Z4 ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G3 ) ) ) ) ).

% isCont_inverse_function
thf(fact_9615_GMVT_H,axiom,
    ! [A2: real,B3: real,F: real > real,G3: real > real,G4: real > real,F6: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ A2 @ Z4 )
           => ( ( ord_less_eq_real @ Z4 @ B3 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A2 @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B3 )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G3 ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_real @ A2 @ Z4 )
               => ( ( ord_less_real @ Z4 @ B3 )
                 => ( has_fi5821293074295781190e_real @ G3 @ ( G4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
           => ( ! [Z4: real] :
                  ( ( ord_less_real @ A2 @ Z4 )
                 => ( ( ord_less_real @ Z4 @ B3 )
                   => ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A2 @ C3 )
                  & ( ord_less_real @ C3 @ B3 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) ) @ ( G4 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G3 @ B3 ) @ ( G3 @ A2 ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9616_summable__Leibniz_I3_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( ( ord_less_real @ ( A2 @ zero_zero_nat ) @ zero_zero_real )
         => ! [N5: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9617_summable__Leibniz_I2_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( ( ord_less_real @ zero_zero_real @ ( A2 @ zero_zero_nat ) )
         => ! [N5: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9618_summable__Leibniz_H_I4_J,axiom,
    ! [A2: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9619_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_9620_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X: nat] : ( times_times_nat @ X @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9621_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_9622_monoseq__convergent,axiom,
    ! [X6: nat > real,B2: real] :
      ( ( topolo6980174941875973593q_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X6 @ I2 ) ) @ B2 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_9623_nested__sequence__unique,axiom,
    ! [F: nat > real,G3: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( G3 @ ( suc @ N3 ) ) @ ( G3 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G3 @ N3 ) )
         => ( ( filterlim_nat_real
              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G3 @ N2 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N5: nat] : ( ord_less_eq_real @ ( F @ N5 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N5: nat] : ( ord_less_eq_real @ L4 @ ( G3 @ N5 ) )
                & ( filterlim_nat_real @ G3 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9624_LIMSEQ__inverse__zero,axiom,
    ! [X6: nat > real] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_real @ R3 @ ( X6 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( X6 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9625_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( root @ N2 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9626_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N5: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9627_LIMSEQ__realpow__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_9628_LIMSEQ__divide__realpow__zero,axiom,
    ! [X2: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( divide_divide_real @ A2 @ ( power_power_real @ X2 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9629_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9630_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9631_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9632_summable,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ) ).

% summable
thf(fact_9633_zeroseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9634_summable__Leibniz_H_I3_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9635_summable__Leibniz_H_I2_J,axiom,
    ! [A2: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9636_sums__alternating__upper__lower,axiom,
    ! [A2: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
       => ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N5: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N5: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9637_summable__Leibniz_H_I5_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9638_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B3: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ B3 @ X5 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_9639_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
            @ at_top_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9640_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B3: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ X5 @ B3 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9641_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
            @ at_bot_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9642_GMVT,axiom,
    ! [A2: real,B3: real,F: real > real,G3: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A2 @ X5 )
              & ( ord_less_eq_real @ X5 @ B3 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ( ! [X5: real] :
              ( ( ( ord_less_real @ A2 @ X5 )
                & ( ord_less_real @ X5 @ B3 ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
         => ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A2 @ X5 )
                  & ( ord_less_eq_real @ X5 @ B3 ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G3 ) )
           => ( ! [X5: real] :
                  ( ( ( ord_less_real @ A2 @ X5 )
                    & ( ord_less_real @ X5 @ B3 ) )
                 => ( differ6690327859849518006l_real @ G3 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G3 @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A2 @ C3 )
                  & ( ord_less_real @ C3 @ B3 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G3 @ B3 ) @ ( G3 @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_9643_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_9644_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_9645_MVT,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A2 @ X5 )
             => ( ( ord_less_real @ X5 @ B3 )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z4: real] :
              ( ( ord_less_real @ A2 @ Z4 )
              & ( ord_less_real @ Z4 @ B3 )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
                = ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_9646_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X5: nat] :
          ( ( ord_less_eq_nat @ C @ X5 )
         => ( P @ X5 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9647_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N4: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N4 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_9648_le__sequentially,axiom,
    ! [F2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
      = ( ! [N4: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N4 ) @ F2 ) ) ) ).

% le_sequentially
thf(fact_9649_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A3 @ F )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A3 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
       => ( topolo5044208981011980120l_real @ A3
          @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_9650_eventually__at__left__real,axiom,
    ! [B3: real,A2: real] :
      ( ( ord_less_real @ B3 @ A2 )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B3 @ A2 ) )
        @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ).

% eventually_at_left_real
thf(fact_9651_continuous__image__closed__interval,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_eq_real @ A2 @ B3 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
       => ? [C3: real,D6: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
              = ( set_or1222579329274155063t_real @ C3 @ D6 ) )
            & ( ord_less_eq_real @ C3 @ D6 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_9652_continuous__on__artanh,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_9653_Rolle__deriv,axiom,
    ! [A2: real,B3: real,F: real > real,F6: real > real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ A2 )
          = ( F @ B3 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A2 @ X5 )
               => ( ( ord_less_real @ X5 @ B3 )
                 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A2 @ Z4 )
                & ( ord_less_real @ Z4 @ B3 )
                & ( ( F6 @ Z4 )
                  = ( ^ [V3: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_9654_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A2: real,B3: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_9655_mvt,axiom,
    ! [A2: real,B3: real,F: real > real,F6: real > real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A2 @ X5 )
             => ( ( ord_less_real @ X5 @ B3 )
               => ( has_de1759254742604945161l_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A2 @ Xi )
               => ( ( ord_less_real @ Xi @ B3 )
                 => ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
                   != ( F6 @ Xi @ ( minus_minus_real @ B3 @ A2 ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_9656_Bseq__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X2 ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_9657_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ A2 @ X5 )
           => ( ( ord_less_real @ X5 @ B3 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
         => ( ord_less_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_9658_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ A2 @ X5 )
           => ( ( ord_less_real @ X5 @ B3 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
         => ( ord_less_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_9659_DERIV__isconst__end,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A2 @ X5 )
             => ( ( ord_less_real @ X5 @ B3 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ( ( F @ B3 )
            = ( F @ A2 ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_9660_DERIV__isconst2,axiom,
    ! [A2: real,B3: real,F: real > real,X2: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A2 @ X5 )
             => ( ( ord_less_real @ X5 @ B3 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A2 @ X2 )
           => ( ( ord_less_eq_real @ X2 @ B3 )
             => ( ( F @ X2 )
                = ( F @ A2 ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_9661_Rolle,axiom,
    ! [A2: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( ( ( F @ A2 )
          = ( F @ B3 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A2 @ X5 )
               => ( ( ord_less_real @ X5 @ B3 )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A2 @ Z4 )
                & ( ord_less_real @ Z4 @ B3 )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_9662_eventually__at__right__real,axiom,
    ! [A2: real,B3: real] :
      ( ( ord_less_real @ A2 @ B3 )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
        @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ).

% eventually_at_right_real
thf(fact_9663_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9664_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_9665_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_9666_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9667_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_9668_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9669_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B3: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9670_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B3 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9671_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9672_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9673_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9674_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_9675_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9676_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_9677_decseq__bounded,axiom,
    ! [X6: nat > real,B2: real] :
      ( ( order_9091379641038594480t_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X6 @ I2 ) )
       => ( bfun_nat_real @ X6 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_9678_decseq__convergent,axiom,
    ! [X6: nat > real,B2: real] :
      ( ( order_9091379641038594480t_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X6 @ I2 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I3: nat] : ( ord_less_eq_real @ L6 @ ( X6 @ I3 ) ) ) ) ) ).

% decseq_convergent
thf(fact_9679_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_9680_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_9681_continuous__on__arcosh,axiom,
    ! [A3: set_real] :
      ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_9682_bdd__above__nat,axiom,
    condit2214826472909112428ve_nat = finite_finite_nat ).

% bdd_above_nat
thf(fact_9683_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y3
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ( Y3
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ 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
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9684_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ~ ( ( 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
                    @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9685_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X2
                = ( 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
                  @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I4: nat] :
                            ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9686_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 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9687_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( 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] :
                ( ( X2
                  = ( 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
                      @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                           => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I4: nat] :
                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9688_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( 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] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( 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
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9689_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y3
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( Y3
                    = ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ 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
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9690_Sup__real__def,axiom,
    ( comple1385675409528146559p_real
    = ( ^ [X8: set_real] :
          ( ord_Least_real
          @ ^ [Z2: real] :
            ! [X: real] :
              ( ( member_real @ X @ X8 )
             => ( ord_less_eq_real @ X @ Z2 ) ) ) ) ) ).

% Sup_real_def
thf(fact_9691_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X8: set_int] :
          ( the_int
          @ ^ [X: int] :
              ( ( member_int @ X @ X8 )
              & ! [Y: int] :
                  ( ( member_int @ Y @ X8 )
                 => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_9692_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_9693_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_9694_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S7: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S7 ) ) @ ( S7 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9695_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S7 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S7 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9696_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N4: nat] :
          ! [M2: nat] :
            ( ( ord_less_eq_nat @ N4 @ M2 )
           => ! [N2: nat] :
                ( ( ord_less_eq_nat @ N4 @ N2 )
               => ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_9697_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_9698_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9699_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9700_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_9701_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_9702_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_9703_incseq__bounded,axiom,
    ! [X6: nat > real,B2: real] :
      ( ( order_mono_nat_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( X6 @ I2 ) @ B2 )
       => ( bfun_nat_real @ X6 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_9704_incseq__convergent,axiom,
    ! [X6: nat > real,B2: real] :
      ( ( order_mono_nat_real @ X6 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( X6 @ I2 ) @ B2 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I3: nat] : ( ord_less_eq_real @ ( X6 @ I3 ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_9705_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S ) ) ) ) ).

% infinite_int_iff_infinite_nat_abs
thf(fact_9706_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_9707_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9708_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G3: nat > nat] :
      ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G3 )
         => ( ( bfun_nat_real
              @ ^ [X: nat] : ( F @ ( G3 @ X ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_9709_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_9710_infinite__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ? [R3: nat > nat] :
          ( ( order_5726023648592871131at_nat @ R3 )
          & ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S ) ) ) ).

% infinite_enumerate
thf(fact_9711_strict__mono__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S ) ) ) ).

% strict_mono_enumerate
thf(fact_9712_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M2: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9713_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9714_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_9715_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_9716_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_9717_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9718_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F6: real > real] :
      ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
     => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F6 @ X5 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_9719_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9720_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9721_log__inj,axiom,
    ! [B3: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( inj_on_real_real @ ( log @ B3 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_9722_inj__Suc,axiom,
    ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).

% inj_Suc
thf(fact_9723_inj__on__diff__nat,axiom,
    ! [N6: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N6 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N6 ) ) ).

% inj_on_diff_nat
thf(fact_9724_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_9725_summable__reindex,axiom,
    ! [F: nat > real,G3: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G3 ) ) ) ) ) ).

% summable_reindex
thf(fact_9726_suminf__reindex__mono,axiom,
    ! [F: nat > real,G3: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G3 ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_9727_suminf__reindex,axiom,
    ! [F: nat > real,G3: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( ! [X5: nat] :
                ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G3 @ top_top_set_nat ) )
               => ( ( F @ X5 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G3 ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_9728_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9729_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_9730_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9731_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9732_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9733_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9734_powr__real__of__int_H,axiom,
    ! [X2: real,N: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( X2 != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N ) )
       => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
          = ( power_int_real @ X2 @ N ) ) ) ) ).

% powr_real_of_int'
thf(fact_9735_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9736_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9737_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9738_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9739_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_9740_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I4: int,N2: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
          & ( N2 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9741_rat__floor__lemma,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A2 @ B3 ) ) @ ( fract @ A2 @ B3 ) )
      & ( ord_less_rat @ ( fract @ A2 @ B3 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9742_less__rat,axiom,
    ! [B3: int,D: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9743_le__rat,axiom,
    ! [B3: int,D: int,A2: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9744_Rats__dense__in__real,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ? [X5: real] :
          ( ( member_real @ X5 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X2 @ X5 )
          & ( ord_less_real @ X5 @ Y3 ) ) ) ).

% Rats_dense_in_real
thf(fact_9745_Rats__no__bot__less,axiom,
    ! [X2: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X5 @ X2 ) ) ).

% Rats_no_bot_less
thf(fact_9746_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q3: rat] :
      ( ! [A: int,B: int] :
          ( ( ord_less_int @ zero_zero_int @ B )
         => ( P @ ( fract @ A @ B ) ) )
     => ( P @ Q3 ) ) ).

% Rat_induct_pos
thf(fact_9747_Rats__no__top__le,axiom,
    ! [X2: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X2 @ X5 ) ) ).

% Rats_no_top_le
thf(fact_9748_zero__less__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A2 @ B3 ) )
        = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).

% zero_less_Fract_iff
thf(fact_9749_Fract__less__zero__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ zero_zero_rat )
        = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_9750_one__less__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A2 @ B3 ) )
        = ( ord_less_int @ B3 @ A2 ) ) ) ).

% one_less_Fract_iff
thf(fact_9751_Fract__less__one__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ one_one_rat )
        = ( ord_less_int @ A2 @ B3 ) ) ) ).

% Fract_less_one_iff
thf(fact_9752_zero__le__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A2 @ B3 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).

% zero_le_Fract_iff
thf(fact_9753_Fract__le__zero__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_9754_one__le__Fract__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A2 @ B3 ) )
        = ( ord_less_eq_int @ B3 @ A2 ) ) ) ).

% one_le_Fract_iff
thf(fact_9755_Fract__le__one__iff,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ one_one_rat )
        = ( ord_less_eq_int @ A2 @ B3 ) ) ) ).

% Fract_le_one_iff
thf(fact_9756_positive__rat,axiom,
    ! [A2: int,B3: int] :
      ( ( positive @ ( fract @ A2 @ B3 ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ).

% positive_rat
thf(fact_9757_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).

% less_rat_def
thf(fact_9758_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_9759_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9760_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9761_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_9762_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_9763_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).

% min_diff
thf(fact_9764_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9765_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_9766_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9767_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
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9768_cis__Arg__unique,axiom,
    ! [Z: complex,X2: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X2 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ( arg @ Z )
            = X2 ) ) ) ) ).

% cis_Arg_unique
thf(fact_9769_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9770_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_9771_bij__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat @ S ) ) ).

% bij_enumerate
thf(fact_9772_Arg__def,axiom,
    ( arg
    = ( ^ [Z2: complex] :
          ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z2 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9773_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9774_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9775_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X2 ) ) ).

% less_eq_int.abs_eq
thf(fact_9776_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9777_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9778_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9779_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X2 ) ) ).

% less_int.abs_eq
thf(fact_9780_card__length__sum__list__rec,axiom,
    ! [M: nat,N6: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N6 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N6 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N6 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9781_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9782_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9783_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_9784_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_9785_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_9786_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9787_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt
thf(fact_9788_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_9789_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast_upt
thf(fact_9790_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_9791_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_9792_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9793_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_9794_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_9795_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9796_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_9797_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_9798_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_9799_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_9800_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X2: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X2 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9801_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_9802_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_9803_sorted__upto,axiom,
    ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).

% sorted_upto
thf(fact_9804_Suc__0__div__numeral,axiom,
    ! [K: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_div_numeral
thf(fact_9805_bezw_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X2 @ Xa2 )
        = Y3 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9806_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9807_bezw__non__0,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y3 )
     => ( ( bezw @ X2 @ Y3 )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9808_bezw_Opelims,axiom,
    ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9809_Suc__0__mod__numeral,axiom,
    ! [K: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9810_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9811_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9812_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
      = ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9813_gcd__ge__0__int,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X2 @ Y3 ) ) ).

% gcd_ge_0_int
thf(fact_9814_gcd__le2__int,axiom,
    ! [B3: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A2 @ B3 ) @ B3 ) ) ).

% gcd_le2_int
thf(fact_9815_gcd__le1__int,axiom,
    ! [A2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A2 @ B3 ) @ A2 ) ) ).

% gcd_le1_int
thf(fact_9816_gcd__cases__int,axiom,
    ! [X2: int,Y3: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
         => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9817_gcd__unique__int,axiom,
    ! [D: int,A2: int,B3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A2 )
        & ( dvd_dvd_int @ D @ B3 )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A2 )
              & ( dvd_dvd_int @ E3 @ B3 ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A2 @ B3 ) ) ) ).

% gcd_unique_int
thf(fact_9818_gcd__non__0__int,axiom,
    ! [Y3: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ Y3 )
     => ( ( gcd_gcd_int @ X2 @ Y3 )
        = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X2 @ Y3 ) ) ) ) ).

% gcd_non_0_int
thf(fact_9819_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( ( gcd_gcd_nat @ A2 @ B3 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9820_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% gcd_nat.left_neutral
thf(fact_9821_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B3: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A2 @ B3 ) )
      = ( ( A2 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9822_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( gcd_gcd_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% gcd_nat.right_neutral
thf(fact_9823_gcd__0__nat,axiom,
    ! [X2: nat] :
      ( ( gcd_gcd_nat @ X2 @ zero_zero_nat )
      = X2 ) ).

% gcd_0_nat
thf(fact_9824_gcd__0__left__nat,axiom,
    ! [X2: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X2 )
      = X2 ) ).

% gcd_0_left_nat
thf(fact_9825_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9826_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
      = ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9827_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A: nat,B: nat] :
          ( ( member_nat @ A @ A3 )
         => ( ( member_nat @ B @ A3 )
           => ( member_nat @ ( gcd_gcd_nat @ A @ B ) @ A3 ) ) )
     => ( ( A3 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).

% Gcd_in
thf(fact_9828_gcd__non__0__nat,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( Y3 != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X2 @ Y3 )
        = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9829_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9830_gcd__nat_Oelims,axiom,
    ! [X2: nat,Xa2: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
        = Y3 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y3 = X2 ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y3
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9831_gcd__le2__nat,axiom,
    ! [B3: nat,A2: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A2 @ B3 ) @ B3 ) ) ).

% gcd_le2_nat
thf(fact_9832_gcd__le1__nat,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A2 @ B3 ) @ A2 ) ) ).

% gcd_le1_nat
thf(fact_9833_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9834_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9835_bezout__nat,axiom,
    ! [A2: nat,B3: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [X5: nat,Y4: nat] :
          ( ( times_times_nat @ A2 @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( gcd_gcd_nat @ A2 @ B3 ) ) ) ) ).

% bezout_nat
thf(fact_9836_bezout__gcd__nat_H,axiom,
    ! [B3: nat,A2: nat] :
    ? [X5: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( times_times_nat @ A2 @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X5 ) @ ( times_times_nat @ B3 @ Y4 ) )
          = ( gcd_gcd_nat @ A2 @ B3 ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ Y4 ) @ ( times_times_nat @ B3 @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B3 @ X5 ) @ ( times_times_nat @ A2 @ Y4 ) )
          = ( gcd_gcd_nat @ A2 @ B3 ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9837_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] :
      ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
      = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).

% Gcd_nat_set_eq_fold
thf(fact_9838_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N2: nat] :
        ( ( dvd_dvd_nat @ M2 @ N2 )
        & ( M2 != N2 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9839_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] :
                ( ( dvd_dvd_nat @ D5 @ M )
                & ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9840_gcd__nat_Opelims,axiom,
    ! [X2: nat,Xa2: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y3 = X2 ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y3
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9841_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( ord_less_nat @ X @ N )
                & ( ord_less_nat @ Y @ N )
                & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
      = ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).

% Field_natLeq_on
thf(fact_9842_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_9843_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_9844_cauchyD,axiom,
    ! [X6: nat > rat,R2: rat] :
      ( ( cauchy @ X6 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ K2 @ M3 )
           => ! [N5: nat] :
                ( ( ord_less_eq_nat @ K2 @ N5 )
               => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M3 ) @ ( X6 @ N5 ) ) ) @ R2 ) ) ) ) ) ).

% cauchyD
thf(fact_9845_cauchy__imp__bounded,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ? [B: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ B )
          & ! [N5: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N5 ) ) @ B ) ) ) ).

% cauchy_imp_bounded
thf(fact_9846_cauchy__def,axiom,
    ( cauchy
    = ( ^ [X8: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [M2: nat] :
              ( ( ord_less_eq_nat @ K3 @ M2 )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).

% cauchy_def
thf(fact_9847_cauchyI,axiom,
    ! [X6: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K7: nat] :
            ! [M4: nat] :
              ( ( ord_less_eq_nat @ K7 @ M4 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K7 @ N3 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) ) @ R3 ) ) ) )
     => ( cauchy @ X6 ) ) ).

% cauchyI
thf(fact_9848_le__Real,axiom,
    ! [X6: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y7 )
       => ( ( ord_less_eq_real @ ( real2 @ X6 ) @ ( real2 @ Y7 ) )
          = ( ! [R5: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ R5 )
               => ? [K3: nat] :
                  ! [N2: nat] :
                    ( ( ord_less_eq_nat @ K3 @ N2 )
                   => ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).

% le_Real
thf(fact_9849_cauchy__not__vanishes,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ? [B: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B )
            & ? [K2: nat] :
              ! [N5: nat] :
                ( ( ord_less_eq_nat @ K2 @ N5 )
               => ( ord_less_rat @ B @ ( abs_abs_rat @ ( X6 @ N5 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes
thf(fact_9850_vanishes__mult__bounded,axiom,
    ! [X6: nat > rat,Y7: nat > rat] :
      ( ? [A9: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ A9 )
          & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N3 ) ) @ A9 ) )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_mult_bounded
thf(fact_9851_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X8: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K3 @ N2 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N2 ) ) @ R5 ) ) ) ) ) ).

% vanishes_def
thf(fact_9852_vanishesI,axiom,
    ! [X6: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K7: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K7 @ N3 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N3 ) ) @ R3 ) ) )
     => ( vanishes @ X6 ) ) ).

% vanishesI
thf(fact_9853_vanishesD,axiom,
    ! [X6: nat > rat,R2: rat] :
      ( ( vanishes @ X6 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K2: nat] :
          ! [N5: nat] :
            ( ( ord_less_eq_nat @ K2 @ N5 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N5 ) ) @ R2 ) ) ) ) ).

% vanishesD
thf(fact_9854_cauchy__not__vanishes__cases,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ? [B: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B )
            & ? [K2: nat] :
                ( ! [N5: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N5 )
                   => ( ord_less_rat @ B @ ( uminus_uminus_rat @ ( X6 @ N5 ) ) ) )
                | ! [N5: nat] :
                    ( ( ord_less_eq_nat @ K2 @ N5 )
                   => ( ord_less_rat @ B @ ( X6 @ N5 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
thf(fact_9855_not__positive__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( ~ ( positive2 @ ( real2 @ X6 ) ) )
        = ( ! [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
             => ? [K3: nat] :
                ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ R5 ) ) ) ) ) ) ).

% not_positive_Real
thf(fact_9856_positive__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( positive2 @ ( real2 @ X6 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ) ).

% positive_Real
thf(fact_9857_less__real__def,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X ) ) ) ) ).

% less_real_def
thf(fact_9858_Real_Opositive_Orep__eq,axiom,
    ( positive2
    = ( ^ [X: real] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K3 @ N2 )
             => ( ord_less_rat @ R5 @ ( rep_real @ X @ N2 ) ) ) ) ) ) ).

% Real.positive.rep_eq
thf(fact_9859_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_9860_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9861_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_9862_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_9863_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_9864_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_9865_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q6: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
              @ ( quotient_of @ Q6 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9866_quotient__of__denom__pos,axiom,
    ! [R2: rat,P6: int,Q3: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P6 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% quotient_of_denom_pos
thf(fact_9867_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9868_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q6: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D5: int] : ( ord_less_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
              @ ( quotient_of @ Q6 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9869_quotient__of__def,axiom,
    ( quotient_of
    = ( ^ [X: rat] :
          ( the_Pr4378521158711661632nt_int
          @ ^ [Pair: product_prod_int_int] :
              ( ( X
                = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).

% quotient_of_def
thf(fact_9870_of__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).

% of_real_sqrt
thf(fact_9871_normalize__stable,axiom,
    ! [Q3: int,P6: int] :
      ( ( ord_less_int @ zero_zero_int @ Q3 )
     => ( ( algebr932160517623751201me_int @ P6 @ Q3 )
       => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
          = ( product_Pair_int_int @ P6 @ Q3 ) ) ) ) ).

% normalize_stable
thf(fact_9872_Rat__induct,axiom,
    ! [P: rat > $o,Q3: rat] :
      ( ! [A: int,B: int] :
          ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( algebr932160517623751201me_int @ A @ B )
           => ( P @ ( fract @ A @ B ) ) ) )
     => ( P @ Q3 ) ) ).

% Rat_induct
thf(fact_9873_Rat__cases,axiom,
    ! [Q3: rat] :
      ~ ! [A: int,B: int] :
          ( ( Q3
            = ( fract @ A @ B ) )
         => ( ( ord_less_int @ zero_zero_int @ B )
           => ~ ( algebr932160517623751201me_int @ A @ B ) ) ) ).

% Rat_cases
thf(fact_9874_Rat__cases__nonzero,axiom,
    ! [Q3: rat] :
      ( ! [A: int,B: int] :
          ( ( Q3
            = ( fract @ A @ B ) )
         => ( ( ord_less_int @ zero_zero_int @ B )
           => ( ( A != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A @ B ) ) ) )
     => ( Q3 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_9875_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X5: product_prod_int_int] :
      ( ( R2
        = ( fract @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) )
      & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X5 ) )
      & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) )
      & ! [Y5: product_prod_int_int] :
          ( ( ( R2
              = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
            & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
            & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
         => ( Y5 = X5 ) ) ) ).

% quotient_of_unique
thf(fact_9876_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_9877_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_9878_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_9879_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_9880_Rats__abs__nat__div__natE,axiom,
    ! [X2: real] :
      ( ( member_real @ X2 @ field_5140801741446780682s_real )
     => ~ ! [M4: nat,N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X2 )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M4 @ N3 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_9881_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9882_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_9883_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_9884_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_9885_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_9886_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_9887_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_9888_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9889_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_9890_Real_Opositive_Oabs__eq,axiom,
    ! [X2: nat > rat] :
      ( ( realrel @ X2 @ X2 )
     => ( ( positive2 @ ( real2 @ X2 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N2 )
                 => ( ord_less_rat @ R5 @ ( X2 @ N2 ) ) ) ) ) ) ) ).

% Real.positive.abs_eq
thf(fact_9891_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9892_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9893_Real_Opositive_Orsp,axiom,
    ( bNF_re728719798268516973at_o_o @ realrel
    @ ^ [Y6: $o,Z3: $o] : Y6 = Z3
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ).

% Real.positive.rsp
thf(fact_9894_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z3: int] : Y6 = Z3
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z3: int] : Y6 = Z3
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_9895_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z3: nat] : Y6 = Z3
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z3: nat] : Y6 = Z3
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9896_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z3: nat] : Y6 = Z3
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z3: nat] : Y6 = Z3
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_9897_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z3: int] : Y6 = Z3
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z3: int] : Y6 = Z3
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_9898_Real_Opositive_Otransfer,axiom,
    ( bNF_re4297313714947099218al_o_o @ pcr_real
    @ ^ [Y6: $o,Z3: $o] : Y6 = Z3
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) )
    @ positive2 ) ).

% Real.positive.transfer
thf(fact_9899_Rat_Opositive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y6: $o,Z3: $o] : Y6 = Z3
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
    @ positive ) ).

% Rat.positive.transfer
thf(fact_9900_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_9901_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_9902_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y6: nat,Z3: nat] : Y6 = Z3
    @ pcr_int
    @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9903_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_9904_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_9905_Rat_Opositive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y6: $o,Z3: $o] : Y6 = Z3
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).

% Rat.positive.rsp
thf(fact_9906_Rat_Opositive_Oabs__eq,axiom,
    ! [X2: product_prod_int_int] :
      ( ( ratrel @ X2 @ X2 )
     => ( ( positive @ ( abs_Rat @ X2 ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).

% Rat.positive.abs_eq
thf(fact_9907_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
      = ( vimage_nat_nat @ suc @ A3 ) ) ).

% vimage_Suc_insert_0
thf(fact_9908_finite__vimage__Suc__iff,axiom,
    ! [F2: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F2 ) )
      = ( finite_finite_nat @ F2 ) ) ).

% finite_vimage_Suc_iff
thf(fact_9909_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_9910_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( ord_less_nat @ X @ N ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N )
              & ( ord_less_nat @ Y @ N )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% Restr_natLeq
thf(fact_9911_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_9912_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
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N )
              & ( ord_less_nat @ Y @ N )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_9913_natLeq__underS__less,axiom,
    ! [N: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
      = ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).

% natLeq_underS_less
thf(fact_9914_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N )
              & ( ord_less_nat @ Y @ N )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_9915_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_9916_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: nat > real,M5: nat] :
      ( ( bfun_nat_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
        @ at_top_nat )
     => ( ! [M4: nat,N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ M4 )
           => ( ( ord_less_eq_nat @ M4 @ N3 )
             => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N3 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_inc
thf(fact_9917_Bseq__mono__convergent,axiom,
    ! [X6: nat > real] :
      ( ( bfun_nat_real @ X6 @ at_top_nat )
     => ( ! [M4: nat,N3: nat] :
            ( ( ord_less_eq_nat @ M4 @ N3 )
           => ( ord_less_eq_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
       => ( topolo7531315842566124627t_real @ X6 ) ) ) ).

% Bseq_mono_convergent
thf(fact_9918_convergent__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( topolo7531315842566124627t_real @ ( power_power_real @ X2 ) ) ) ) ).

% convergent_realpow
thf(fact_9919_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N ) ) ).

% less_eq
thf(fact_9920_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: nat > real,M5: nat] :
      ( ( bfun_nat_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
        @ at_top_nat )
     => ( ! [M4: nat,N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ M4 )
           => ( ( ord_less_eq_nat @ M4 @ N3 )
             => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ M4 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_dec
thf(fact_9921_pair__lessI2,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_9922_trans__pair__less,axiom,
    trans_4347625901269045472at_nat @ fun_pair_less ).

% trans_pair_less
thf(fact_9923_total__pair__less,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A3 @ fun_pair_less ) ).

% total_pair_less
thf(fact_9924_pair__less__iff1,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( product_Pair_nat_nat @ X2 @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z ) ) ).

% pair_less_iff1
thf(fact_9925_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_9926_pair__lessI1,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_9927_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N2: nat] :
        ( ( dvd_dvd_nat @ M2 @ N2 )
        & ( M2 != N2 ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_9928_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_9929_pair__leqI2,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A2 @ B3 )
     => ( ( ord_less_eq_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_9930_pair__leqI1,axiom,
    ! [A2: nat,B3: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_9931_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_9932_wmin__insertI,axiom,
    ! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_9933_wmax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_9934_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_9935_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X6 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_9936_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_9937_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_9938_smin__insertI,axiom,
    ! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X2 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_9939_smax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ X6 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_9940_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_9941_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_9942_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_9943_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_9944_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_9945_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_9946_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9947_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_9948_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9949_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9950_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_9951_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_9952_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_9953_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).

% less_int.rsp
thf(fact_9954_division__segment__int__def,axiom,
    ( euclid3395696857347342551nt_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% division_segment_int_def
thf(fact_9955_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M2: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M2 )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_9956_less__enat__def,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [M2: extended_enat,N2: extended_enat] :
          ( extended_case_enat_o
          @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
          @ $false
          @ M2 ) ) ) ).

% less_enat_def
thf(fact_9957_of__rat__dense,axiom,
    ! [X2: real,Y3: real] :
      ( ( ord_less_real @ X2 @ Y3 )
     => ? [Q4: rat] :
          ( ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ Q4 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q4 ) @ Y3 ) ) ) ).

% of_rat_dense
thf(fact_9958_less__RealD,axiom,
    ! [Y7: nat > rat,X2: real] :
      ( ( cauchy @ Y7 )
     => ( ( ord_less_real @ X2 @ ( real2 @ Y7 ) )
       => ? [N3: nat] : ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ ( Y7 @ N3 ) ) ) ) ) ).

% less_RealD
thf(fact_9959_Real__leI,axiom,
    ! [X6: nat > rat,Y3: real] :
      ( ( cauchy @ X6 )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X6 @ N3 ) ) @ Y3 )
       => ( ord_less_eq_real @ ( real2 @ X6 ) @ Y3 ) ) ) ).

% Real_leI
thf(fact_9960_le__RealI,axiom,
    ! [Y7: nat > rat,X2: real] :
      ( ( cauchy @ Y7 )
     => ( ! [N3: nat] : ( ord_less_eq_real @ X2 @ ( field_7254667332652039916t_real @ ( Y7 @ N3 ) ) )
       => ( ord_less_eq_real @ X2 @ ( real2 @ Y7 ) ) ) ) ).

% le_RealI
thf(fact_9961_compute__powr__real,axiom,
    ( powr_real2
    = ( ^ [B4: real,I4: real] :
          ( if_real @ ( ord_less_eq_real @ B4 @ zero_zero_real )
          @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) )
          @ ( if_real
            @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
              = I4 )
            @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
            @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) ) ) ) ) ) ).

% compute_powr_real
thf(fact_9962_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_9963_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_9964_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_9965_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_9966_numeral__le__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% numeral_le_enat_iff
thf(fact_9967_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% enat_ord_simps(2)
thf(fact_9968_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% enat_ord_simps(1)
thf(fact_9969_idiff__enat__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
      = N ) ).

% idiff_enat_0_right
thf(fact_9970_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_9971_numeral__less__enat__iff,axiom,
    ! [M: num,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% numeral_less_enat_iff
thf(fact_9972_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_9973_enat__0__iff_I1_J,axiom,
    ! [X2: nat] :
      ( ( ( extended_enat2 @ X2 )
        = zero_z5237406670263579293d_enat )
      = ( X2 = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_9974_enat__0__iff_I2_J,axiom,
    ! [X2: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X2 ) )
      = ( X2 = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_9975_finite__enat__bounded,axiom,
    ! [A3: set_Extended_enat,N: nat] :
      ( ! [Y4: extended_enat] :
          ( ( member_Extended_enat @ Y4 @ A3 )
         => ( ord_le2932123472753598470d_enat @ Y4 @ ( extended_enat2 @ N ) ) )
     => ( finite4001608067531595151d_enat @ A3 ) ) ).

% finite_enat_bounded
thf(fact_9976_enat__ile,axiom,
    ! [N: extended_enat,M: nat] :
      ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
     => ? [K2: nat] :
          ( N
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_ile
thf(fact_9977_enat__iless,axiom,
    ! [N: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
     => ? [K2: nat] :
          ( N
          = ( extended_enat2 @ K2 ) ) ) ).

% enat_iless
thf(fact_9978_less__enatE,axiom,
    ! [N: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
     => ~ ! [K2: nat] :
            ( ( N
              = ( extended_enat2 @ K2 ) )
           => ~ ( ord_less_nat @ K2 @ M ) ) ) ).

% less_enatE
thf(fact_9979_Suc__ile__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).

% Suc_ile_eq
thf(fact_9980_iadd__le__enat__iff,axiom,
    ! [X2: extended_enat,Y3: extended_enat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ ( extended_enat2 @ N ) )
      = ( ? [Y8: nat,X9: nat] :
            ( ( X2
              = ( extended_enat2 @ X9 ) )
            & ( Y3
              = ( extended_enat2 @ Y8 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_9981_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_9982_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_9983_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_9984_enat__ord__simps_I4_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Q3 @ extend5688581933313929465d_enat )
      = ( Q3 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
thf(fact_9985_enat__ord__simps_I6_J,axiom,
    ! [Q3: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q3 ) ).

% enat_ord_simps(6)
thf(fact_9986_enat__ord__code_I3_J,axiom,
    ! [Q3: extended_enat] : ( ord_le2932123472753598470d_enat @ Q3 @ extend5688581933313929465d_enat ) ).

% enat_ord_code(3)
thf(fact_9987_enat__ord__simps_I5_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q3 )
      = ( Q3 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
thf(fact_9988_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      ( ( ( M = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
          = zero_z5237406670263579293d_enat ) )
      & ( ( M != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
thf(fact_9989_enat__ord__code_I5_J,axiom,
    ! [N: nat] :
      ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).

% enat_ord_code(5)
thf(fact_9990_infinity__ileE,axiom,
    ! [M: nat] :
      ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).

% infinity_ileE
thf(fact_9991_enat__add__left__cancel__less,axiom,
    ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B3 ) @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
      = ( ( A2 != extend5688581933313929465d_enat )
        & ( ord_le72135733267957522d_enat @ B3 @ C ) ) ) ).

% enat_add_left_cancel_less
thf(fact_9992_enat__ord__code_I4_J,axiom,
    ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).

% enat_ord_code(4)
thf(fact_9993_less__infinityE,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ N @ extend5688581933313929465d_enat )
     => ~ ! [K2: nat] :
            ( N
           != ( extended_enat2 @ K2 ) ) ) ).

% less_infinityE
thf(fact_9994_infinity__ilessE,axiom,
    ! [M: nat] :
      ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).

% infinity_ilessE
thf(fact_9995_enat__add__left__cancel__le,axiom,
    ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B3 ) @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
      = ( ( A2 = extend5688581933313929465d_enat )
        | ( ord_le2932123472753598470d_enat @ B3 @ C ) ) ) ).

% enat_add_left_cancel_le
thf(fact_9996_enat__ord__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ( ord_le2932123472753598470d_enat @ Q3 @ extend5688581933313929465d_enat ) ).

% enat_ord_simps(3)
thf(fact_9997_Inf__enat__def,axiom,
    ( comple2295165028678016749d_enat
    = ( ^ [A6: set_Extended_enat] :
          ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
          @ ( ord_Le1955565732374568822d_enat
            @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 ) ) ) ) ) ).

% Inf_enat_def
thf(fact_9998_Sup__enat__def,axiom,
    ( comple4398354569131411667d_enat
    = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).

% Sup_enat_def
thf(fact_9999_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_10000_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_10001_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M2: extended_enat,N2: 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 )
              @ N2 )
          @ ( if_Extended_enat @ ( N2 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M2 ) ) ) ).

% times_enat_def
thf(fact_10002_iless__Suc__eq,axiom,
    ! [M: nat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
      = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N ) ) ).

% iless_Suc_eq
thf(fact_10003_eSuc__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
      = ( ord_le72135733267957522d_enat @ N @ M ) ) ).

% eSuc_mono
thf(fact_10004_eSuc__ile__mono,axiom,
    ! [N: extended_enat,M: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
      = ( ord_le2932123472753598470d_enat @ N @ M ) ) ).

% eSuc_ile_mono
thf(fact_10005_iless__eSuc0,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ N @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
thf(fact_10006_eSuc__Max,axiom,
    ! [A3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A3 )
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A3 ) )
          = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ) ).

% eSuc_Max
thf(fact_10007_eSuc__Sup,axiom,
    ! [A3: set_Extended_enat] :
      ( ( A3 != bot_bo7653980558646680370d_enat )
     => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A3 ) )
        = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ).

% eSuc_Sup
thf(fact_10008_not__eSuc__ilei0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_eSuc_ilei0
thf(fact_10009_ile__eSuc,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).

% ile_eSuc
thf(fact_10010_ileI1,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M @ N )
     => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N ) ) ).

% ileI1
thf(fact_10011_i0__iless__eSuc,axiom,
    ! [N: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N ) ) ).

% i0_iless_eSuc
thf(fact_10012_less__than__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ less_than )
      = ( ord_less_nat @ X2 @ Y3 ) ) ).

% less_than_iff
thf(fact_10013_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_10014_natLeq__on__well__order__on,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N )
              & ( ord_less_nat @ Y @ N )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_10015_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( ord_less_nat @ X @ N )
                & ( ord_less_nat @ Y @ N )
                & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N )
              & ( ord_less_nat @ Y @ N )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_10016_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_10017_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_10018_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_10019_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_10020_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_10021_diff__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N ) ) ) ).

% diff_Suc
thf(fact_10022_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_10023_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_10024_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_10025_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_10026_Rep__unit__induct,axiom,
    ! [Y3: $o,P: $o > $o] :
      ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X5: product_unit] : ( P @ ( product_Rep_unit @ X5 ) )
       => ( P @ Y3 ) ) ) ).

% Rep_unit_induct
thf(fact_10027_Abs__unit__inject,axiom,
    ! [X2: $o,Y3: $o] :
      ( ( member_o @ X2 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X2 )
            = ( product_Abs_unit @ Y3 ) )
          = ( X2 = Y3 ) ) ) ) ).

% Abs_unit_inject
thf(fact_10028_Abs__unit__inverse,axiom,
    ! [Y3: $o] :
      ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
        = Y3 ) ) ).

% Abs_unit_inverse
thf(fact_10029_Rep__unit,axiom,
    ! [X2: product_unit] : ( member_o @ ( product_Rep_unit @ X2 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_10030_Abs__unit__cases,axiom,
    ! [X2: product_unit] :
      ~ ! [Y4: $o] :
          ( ( X2
            = ( product_Abs_unit @ Y4 ) )
         => ~ ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_10031_Rep__unit__cases,axiom,
    ! [Y3: $o] :
      ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X5: product_unit] :
            ( Y3
            = ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).

% Rep_unit_cases
thf(fact_10032_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X2: product_unit] :
      ( ! [Y4: $o] :
          ( ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y4 ) ) )
     => ( P @ X2 ) ) ).

% Abs_unit_induct
thf(fact_10033_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_10034_Real_Opositive__def,axiom,
    ( positive2
    = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
      @ ^ [X8: nat > rat] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K3 @ N2 )
             => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ).

% Real.positive_def
thf(fact_10035_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive_def
thf(fact_10036_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_10037_complex__Re__le__cmod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( re @ X2 ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% complex_Re_le_cmod
thf(fact_10038_abs__Re__le__cmod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% abs_Re_le_cmod
thf(fact_10039_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_10040_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_10041_csqrt__unique,axiom,
    ! [W2: complex,Z: complex] :
      ( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
          | ( ( ( re @ W2 )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
       => ( ( csqrt @ Z )
          = W2 ) ) ) ).

% csqrt_unique
thf(fact_10042_csqrt__of__real__nonneg,axiom,
    ! [X2: complex] :
      ( ( ( im @ X2 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) )
       => ( ( csqrt @ X2 )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X2 ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_10043_abs__Im__le__cmod,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% abs_Im_le_cmod
thf(fact_10044_cmod__Im__le__iff,axiom,
    ! [X2: complex,Y3: complex] :
      ( ( ( re @ X2 )
        = ( re @ Y3 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_10045_cmod__Re__le__iff,axiom,
    ! [X2: complex,Y3: complex] :
      ( ( ( im @ X2 )
        = ( im @ Y3 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_10046_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_10047_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_10048_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_10049_csqrt__square,axiom,
    ! [B3: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B3 ) )
        | ( ( ( re @ B3 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B3 ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B3 ) ) ).

% csqrt_square
thf(fact_10050_csqrt__of__real__nonpos,axiom,
    ! [X2: complex] :
      ( ( ( im @ X2 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
       => ( ( csqrt @ X2 )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_10051_csqrt__minus,axiom,
    ! [X2: complex] :
      ( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
        | ( ( ( im @ X2 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).

% csqrt_minus
thf(fact_10052_complex__div__gt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_10053_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_10054_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_10055_Re__complex__div__le__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_10056_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_10057_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_10058_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_10059_Im__complex__div__le__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_10060_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B3: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_10061_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_10062_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).

% less_int_def
thf(fact_10063_MOST__SucD,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
        @ cofinite_nat )
     => ( eventually_nat @ P @ cofinite_nat ) ) ).

% MOST_SucD
thf(fact_10064_MOST__SucI,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
     => ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
        @ cofinite_nat ) ) ).

% MOST_SucI
thf(fact_10065_MOST__Suc__iff,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
        @ cofinite_nat )
      = ( eventually_nat @ P @ cofinite_nat ) ) ).

% MOST_Suc_iff
thf(fact_10066_MOST__nat__le,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
      = ( ? [M2: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% MOST_nat_le
thf(fact_10067_MOST__ge__nat,axiom,
    ! [M: nat] : ( eventually_nat @ ( ord_less_eq_nat @ M ) @ cofinite_nat ) ).

% MOST_ge_nat
thf(fact_10068_MOST__nat,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ cofinite_nat )
      = ( ? [M2: nat] :
          ! [N2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% MOST_nat
thf(fact_10069_infinity__enat__def,axiom,
    ( extend5688581933313929465d_enat
    = ( extended_Abs_enat @ none_nat ) ) ).

% infinity_enat_def
thf(fact_10070_enat__def,axiom,
    ( extended_enat2
    = ( ^ [N2: nat] : ( extended_Abs_enat @ ( some_nat @ N2 ) ) ) ) ).

% enat_def
thf(fact_10071_INFM__nat__le,axiom,
    ! [P: nat > $o] :
      ( ( frequently_nat @ P @ cofinite_nat )
      = ( ! [M2: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
            & ( P @ N2 ) ) ) ) ).

% INFM_nat_le
thf(fact_10072_INFM__nat,axiom,
    ! [P: nat > $o] :
      ( ( frequently_nat @ P @ cofinite_nat )
      = ( ! [M2: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
            & ( P @ N2 ) ) ) ) ).

% INFM_nat
thf(fact_10073_list__encode_Oelims,axiom,
    ! [X2: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X2 )
        = Y3 )
     => ( ( ( X2 = nil_nat )
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs3: list_nat] :
              ( ( X2
                = ( cons_nat @ X5 @ Xs3 ) )
             => ( Y3
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_10074_le__prod__encode__1,axiom,
    ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ).

% le_prod_encode_1
thf(fact_10075_le__prod__encode__2,axiom,
    ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ).

% le_prod_encode_2
thf(fact_10076_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_10077_list__encode_Opelims,axiom,
    ! [X2: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X2 )
        = Y3 )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X2 )
       => ( ( ( X2 = nil_nat )
           => ( ( Y3 = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X5: nat,Xs3: list_nat] :
                ( ( X2
                  = ( cons_nat @ X5 @ Xs3 ) )
               => ( ( Y3
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_10078_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_10079_list__decode_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N3: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
             => ( ! [X4: nat,Y5: nat] :
                    ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
                      = ( nat_prod_decode @ N3 ) )
                   => ( P @ Y5 ) )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_10080_list__decode_Oelims,axiom,
    ! [X2: nat,Y3: list_nat] :
      ( ( ( nat_list_decode @ X2 )
        = Y3 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y3 != nil_nat ) )
       => ~ ! [N3: nat] :
              ( ( X2
                = ( suc @ N3 ) )
             => ( Y3
               != ( produc2761476792215241774st_nat
                  @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
                  @ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_10081_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_10082_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_10083_list__decode_Opelims,axiom,
    ! [X2: nat,Y3: list_nat] :
      ( ( ( nat_list_decode @ X2 )
        = Y3 )
     => ( ( accp_nat @ nat_list_decode_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y3 = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N3: nat] :
                ( ( X2
                  = ( suc @ N3 ) )
               => ( ( Y3
                    = ( produc2761476792215241774st_nat
                      @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
                      @ ( nat_prod_decode @ N3 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_10084_unit__factor__simps_I1_J,axiom,
    ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% unit_factor_simps(1)
thf(fact_10085_unit__factor__nat__def,axiom,
    ( unit_f2748546683901255202or_nat
    = ( ^ [N2: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).

% unit_factor_nat_def
thf(fact_10086_Lcm__eq__0__I__nat,axiom,
    ! [A3: set_nat] :
      ( ( member_nat @ zero_zero_nat @ A3 )
     => ( ( gcd_Lcm_nat @ A3 )
        = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
thf(fact_10087_Lcm__0__iff__nat,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( gcd_Lcm_nat @ A3 )
          = zero_zero_nat )
        = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).

% Lcm_0_iff_nat
thf(fact_10088_Lcm__nat__infinite,axiom,
    ! [M5: set_nat] :
      ( ~ ( finite_finite_nat @ M5 )
     => ( ( gcd_Lcm_nat @ M5 )
        = zero_zero_nat ) ) ).

% Lcm_nat_infinite
thf(fact_10089_Lcm__int__greater__eq__0,axiom,
    ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K4 ) ) ).

% Lcm_int_greater_eq_0
thf(fact_10090_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_10091_Lcm__eq__Max__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ! [M4: nat,N3: nat] :
                ( ( member_nat @ M4 @ M5 )
               => ( ( member_nat @ N3 @ M5 )
                 => ( member_nat @ ( gcd_lcm_nat @ M4 @ N3 ) @ M5 ) ) )
           => ( ( gcd_Lcm_nat @ M5 )
              = ( lattic8265883725875713057ax_nat @ M5 ) ) ) ) ) ) ).

% Lcm_eq_Max_nat
thf(fact_10092_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ( gcd_lcm_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
thf(fact_10093_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ( gcd_lcm_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% lcm_1_iff_nat
thf(fact_10094_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ! [M4: nat,N3: nat] :
              ( ( member_nat @ M4 @ M5 )
             => ( ( member_nat @ N3 @ M5 )
               => ( member_nat @ ( gcd_lcm_nat @ M4 @ N3 ) @ M5 ) ) )
         => ( member_nat @ ( gcd_Lcm_nat @ M5 ) @ M5 ) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
thf(fact_10095_lcm__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( N != zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).

% lcm_pos_int
thf(fact_10096_lcm__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).

% lcm_pos_nat
thf(fact_10097_lcm__ge__0__int,axiom,
    ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X2 @ Y3 ) ) ).

% lcm_ge_0_int
thf(fact_10098_lcm__unique__int,axiom,
    ! [D: int,A2: int,B3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ A2 @ D )
        & ( dvd_dvd_int @ B3 @ D )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ A2 @ E3 )
              & ( dvd_dvd_int @ B3 @ E3 ) )
           => ( dvd_dvd_int @ D @ E3 ) ) )
      = ( D
        = ( gcd_lcm_int @ A2 @ B3 ) ) ) ).

% lcm_unique_int
thf(fact_10099_lcm__cases__int,axiom,
    ! [X2: int,Y3: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( P @ ( gcd_lcm_int @ X2 @ Y3 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
           => ( P @ ( gcd_lcm_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
             => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
         => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
               => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
           => ( P @ ( gcd_lcm_int @ X2 @ Y3 ) ) ) ) ) ) ).

% lcm_cases_int
thf(fact_10100_Lcm__nat__def,axiom,
    ( gcd_Lcm_nat
    = ( ^ [M8: set_nat] : ( if_nat @ ( finite_finite_nat @ M8 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M8 ) @ zero_zero_nat ) ) ) ).

% Lcm_nat_def
thf(fact_10101_VEBT__internal_Olesseq_Osimps,axiom,
    ( vEBT_VEBT_lesseq
    = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat ) ) ).

% VEBT_internal.lesseq.simps
thf(fact_10102_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
    ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
      ( ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
        = Y3 )
     => ( Y3
        = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ).

% VEBT_internal.lesseq.elims(1)
thf(fact_10103_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
     => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ).

% VEBT_internal.lesseq.elims(2)
thf(fact_10104_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
     => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ).

% VEBT_internal.lesseq.elims(3)
thf(fact_10105_VEBT__internal_Oless_Oelims_I3_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_less @ X2 @ Xa2 )
     => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ).

% VEBT_internal.less.elims(3)
thf(fact_10106_VEBT__internal_Oless_Oelims_I2_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_less @ X2 @ Xa2 )
     => ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ).

% VEBT_internal.less.elims(2)
thf(fact_10107_VEBT__internal_Oless_Oelims_I1_J,axiom,
    ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
      ( ( ( vEBT_VEBT_less @ X2 @ Xa2 )
        = Y3 )
     => ( Y3
        = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ).

% VEBT_internal.less.elims(1)
thf(fact_10108_VEBT__internal_Oless_Osimps,axiom,
    ( vEBT_VEBT_less
    = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat ) ) ).

% VEBT_internal.less.simps
thf(fact_10109_VEBT__internal_Ogreater_Osimps,axiom,
    ( vEBT_VEBT_greater
    = ( vEBT_V2881884560877996034ft_nat
      @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ) ).

% VEBT_internal.greater.simps
thf(fact_10110_VEBT__internal_Ogreater_Oelims_I1_J,axiom,
    ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
      ( ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
        = Y3 )
     => ( Y3
        = ( vEBT_V2881884560877996034ft_nat
          @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
          @ X2
          @ Xa2 ) ) ) ).

% VEBT_internal.greater.elims(1)
thf(fact_10111_VEBT__internal_Ogreater_Oelims_I2_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
     => ( vEBT_V2881884560877996034ft_nat
        @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
        @ X2
        @ Xa2 ) ) ).

% VEBT_internal.greater.elims(2)
thf(fact_10112_VEBT__internal_Ogreater_Oelims_I3_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_greater @ X2 @ Xa2 )
     => ~ ( vEBT_V2881884560877996034ft_nat
          @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
          @ X2
          @ Xa2 ) ) ).

% VEBT_internal.greater.elims(3)
thf(fact_10113_VEBT__internal_Ogreater_Opelims_I1_J,axiom,
    ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
      ( ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( Y3
              = ( vEBT_V2881884560877996034ft_nat
                @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
                @ X2
                @ Xa2 ) )
           => ~ ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).

% VEBT_internal.greater.pelims(1)
thf(fact_10114_VEBT__internal_Ogreater_Ocases,axiom,
    ! [X2: produc4953844613479565601on_nat] :
      ~ ! [X5: option_nat,Y4: option_nat] :
          ( X2
         != ( produc5098337634421038937on_nat @ X5 @ Y4 ) ) ).

% VEBT_internal.greater.cases
thf(fact_10115_VEBT__internal_Ogreater_Opelims_I3_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_greater @ X2 @ Xa2 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
           => ( vEBT_V2881884560877996034ft_nat
              @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
              @ X2
              @ Xa2 ) ) ) ) ).

% VEBT_internal.greater.pelims(3)
thf(fact_10116_VEBT__internal_Ogreater_Opelims_I2_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
           => ~ ( vEBT_V2881884560877996034ft_nat
                @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
                @ X2
                @ Xa2 ) ) ) ) ).

% VEBT_internal.greater.pelims(2)
thf(fact_10117_VEBT__internal_Olesseq_Opelims_I1_J,axiom,
    ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
      ( ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( Y3
              = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) )
           => ~ ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).

% VEBT_internal.lesseq.pelims(1)
thf(fact_10118_VEBT__internal_Olesseq_Opelims_I2_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
           => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ) ).

% VEBT_internal.lesseq.pelims(2)
thf(fact_10119_VEBT__internal_Olesseq_Opelims_I3_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
           => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ) ).

% VEBT_internal.lesseq.pelims(3)
thf(fact_10120_VEBT__internal_Oless_Opelims_I3_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_less @ X2 @ Xa2 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
           => ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ) ).

% VEBT_internal.less.pelims(3)
thf(fact_10121_VEBT__internal_Oless_Opelims_I2_J,axiom,
    ! [X2: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_less @ X2 @ Xa2 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
           => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ) ).

% VEBT_internal.less.pelims(2)
thf(fact_10122_VEBT__internal_Oless_Opelims_I1_J,axiom,
    ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
      ( ( ( vEBT_VEBT_less @ X2 @ Xa2 )
        = Y3 )
     => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
       => ~ ( ( Y3
              = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) )
           => ~ ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).

% VEBT_internal.less.pelims(1)

% Helper facts (38)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y3: int] :
      ( ( if_int @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y3: int] :
      ( ( if_int @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( if_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( if_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X2: num,Y3: num] :
      ( ( if_num @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X2: num,Y3: num] :
      ( ( if_num @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( if_rat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y3: rat] :
      ( ( if_rat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y3: real] :
      ( ( if_real @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y3: real] :
      ( ( if_real @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X8: real] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y3: complex] :
      ( ( if_complex @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y3: complex] :
      ( ( if_complex @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( if_set_int @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X2: set_int,Y3: set_int] :
      ( ( if_set_int @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y3: list_int] :
      ( ( if_list_int @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y3: list_int] :
      ( ( if_list_int @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y3: option_nat] :
      ( ( if_option_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y3: option_nat] :
      ( ( if_option_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X2: option_num,Y3: option_num] :
      ( ( if_option_num @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X2: option_num,Y3: option_num] :
      ( ( if_option_num @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (4)
thf(conj_0,hypothesis,
    ~ ( ( ( vEBT_vebt_mint @ ta )
        = none_nat )
      & ( ( vEBT_vebt_mint @ k )
        = ( some_nat @ b ) ) ) ).

thf(conj_1,hypothesis,
    ~ ( ( ( vEBT_vebt_mint @ ta )
        = ( some_nat @ a ) )
      & ( ( vEBT_vebt_mint @ k )
        = none_nat ) ) ).

thf(conj_2,hypothesis,
    ( ( ord_less_nat @ a @ b )
    & ( ( some_nat @ a )
      = ( vEBT_vebt_mint @ ta ) )
    & ( ( some_nat @ b )
      = ( vEBT_vebt_mint @ k ) ) ) ).

thf(conj_3,conjecture,
    $false ).

%------------------------------------------------------------------------------