TPTP Problem File: ITP233^1.p

%------------------------------------------------------------------------------
% File     : ITP233^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 00603_038671
% 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    : 0067_VEBT_InsertCorrectness_00603_038671 [Des22]

% Status   : Theorem
% Rating   : 0.60 v8.2.0, 0.38 v8.1.0
% Syntax   : Number of formulae    : 10382 (4586 unt;1264 typ;   0 def)
%            Number of atoms       : 26612 (11206 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 102627 (2843   ~; 440   |;1963   &;86410   @)
%                                         (   0 <=>;10971  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  144 ( 143 usr)
%            Number of type conns  : 5246 (5246   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1124 (1121 usr;  69 con; 0-5 aty)
%            Number of variables   : 24747 (2389   ^;21656   !; 702   ?;24747   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 21:14:47.668
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
    option_real: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
    list_num: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

thf(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $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__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__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__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 (1121)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > 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__Real__Oreal,type,
    archim2898591450579166408c_real: real > real ).

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_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__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Int__Oint_001t__Int__Oint,type,
    basic_1872990034501187214nt_int: ( int > nat ) > ( int > nat ) > product_prod_int_int > nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    basic_876126793109182934at_nat: ( nat > nat ) > ( nat > nat ) > product_prod_nat_nat > nat ).

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__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_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_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

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

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

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

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

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

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

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__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__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_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__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__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

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

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

thf(sy_c_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_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__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__Int__Oint,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

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

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

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

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

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
    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_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Extended____Nat__Oenat,type,
    comm_s3181272606743183617d_enat: extended_enat > nat > extended_enat ).

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

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

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Extended____Nat__Oenat,type,
    semiri4449623510593786356d_enat: nat > extended_enat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
    semiri1406184849735516958ct_int: nat > int ).

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__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

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_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,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

thf(sy_c_Finite__Set_Ocard_001_Eo,type,
    finite_card_o: set_o > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Extended____Nat__Oenat,type,
    finite121521170596916366d_enat: set_Extended_enat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
    finite_card_int: set_int > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite5120063068150530198omplex: set_list_complex > nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Ocan__select_001_Eo,type,
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thf(sy_c_List_Ocan__select_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Oconcat_001t__Int__Oint,type,
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thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
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thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
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thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Odistinct_001_Eo,type,
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thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
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thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ofind_001_Eo,type,
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thf(sy_c_List_Ofind_001t__Int__Oint,type,
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thf(sy_c_List_Ofind_001t__Nat__Onat,type,
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thf(sy_c_List_Ofind_001t__Num__Onum,type,
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thf(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Oinsert_001_Eo,type,
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thf(sy_c_List_Oinsert_001t__Int__Oint,type,
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thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_List_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olast_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
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thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
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thf(sy_c_List_Olist_Osize__list_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Osize__list_001t__Real__Oreal,type,
    size_list_real: ( real > nat ) > list_real > nat ).

thf(sy_c_List_Olist_Osize__list_001t__Set__Oset_It__Nat__Onat_J,type,
    size_list_set_nat: ( set_nat > nat ) > list_set_nat > nat ).

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

thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

thf(sy_c_List_Olist__ex1_001_Eo,type,
    list_ex1_o: ( $o > $o ) > list_o > $o ).

thf(sy_c_List_Olist__ex1_001t__Int__Oint,type,
    list_ex1_int: ( int > $o ) > list_int > $o ).

thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
    list_ex1_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
    list_ex1_real: ( real > $o ) > list_real > $o ).

thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__Nat__Onat_J,type,
    list_ex1_set_nat: ( set_nat > $o ) > list_set_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__VEBT____Definitions__OVEBT,type,
    list_ex1_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > $o ).

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__Extended____Nat__Oenat,type,
    list_u3071683517702156500d_enat: list_Extended_enat > nat > extended_enat > list_Extended_enat ).

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__Int__Oint_Mt__Int__Oint_J,type,
    list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_u5003261594476800725at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    list_u2241885204319820103d_enat: list_P6356568628958627295d_enat > nat > produc7272778201969148633d_enat > list_P6356568628958627295d_enat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    list_u2459188882655168453BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    list_u6961636818849549845T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT > list_P7413028617227757229T_VEBT ).

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_Olistrel1_001t__Int__Oint,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    listre4828114922151135584at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).

thf(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
    listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(sy_c_List_Olistrel1p_001t__Int__Oint,type,
    listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).

thf(sy_c_List_Olistrel1p_001t__Nat__Onat,type,
    listrel1p_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_List_Omember_001_Eo,type,
    member_o: list_o > $o > $o ).

thf(sy_c_List_Omember_001t__Int__Oint,type,
    member_int: list_int > int > $o ).

thf(sy_c_List_Omember_001t__Nat__Onat,type,
    member_nat: list_nat > nat > $o ).

thf(sy_c_List_Omember_001t__Real__Oreal,type,
    member_real: list_real > real > $o ).

thf(sy_c_List_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: list_set_nat > set_nat > $o ).

thf(sy_c_List_Omember_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_List_On__lists_001t__Int__Oint,type,
    n_lists_int: nat > list_int > list_list_int ).

thf(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_On__lists_001t__VEBT____Definitions__OVEBT,type,
    n_lists_VEBT_VEBT: nat > list_VEBT_VEBT > list_list_VEBT_VEBT ).

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

thf(sy_c_List_Onth_001t__Extended____Nat__Oenat,type,
    nth_Extended_enat: list_Extended_enat > nat > extended_enat ).

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_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_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_Mt__Extended____Nat__Oenat_J,type,
    nth_Pr7509392720524132704d_enat: list_P6356568628958627295d_enat > nat > produc7272778201969148633d_enat ).

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_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

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

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

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
    product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

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

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

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc4894985897562121079d_enat: list_VEBT_VEBT > list_Extended_enat > list_P6356568628958627295d_enat ).

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_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

thf(sy_c_List_Oremove1_001_Eo,type,
    remove1_o: $o > list_o > list_o ).

thf(sy_c_List_Oremove1_001t__Int__Oint,type,
    remove1_int: int > list_int > list_int ).

thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
    remove1_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oremove1_001t__Real__Oreal,type,
    remove1_real: real > list_real > list_real ).

thf(sy_c_List_Oremove1_001t__Set__Oset_It__Nat__Onat_J,type,
    remove1_set_nat: set_nat > list_set_nat > list_set_nat ).

thf(sy_c_List_Oremove1_001t__VEBT____Definitions__OVEBT,type,
    remove1_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

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

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

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

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

thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: nat > set_nat > list_set_nat ).

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

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_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
    subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).

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

thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).

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

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

thf(sy_c_List_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_001_Eo,type,
    zip_o_o: list_o > list_o > list_P4002435161011370285od_o_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__Nat__Onat,type,
    zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
    zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).

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__Real__Oreal,type,
    zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).

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__Nat__Onat_001t__Int__Oint,type,
    zip_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
    zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).

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__Extended____Nat__Oenat,type,
    zip_VE7205001627739651817d_enat: list_VEBT_VEBT > list_Extended_enat > list_P6356568628958627295d_enat ).

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__Real__Oreal,type,
    zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

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__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    compow6801810373992395016omplex: nat > ( complex > complex ) > complex > complex ).

thf(sy_c_Nat_Ocompow_001_062_It__Extended____Nat__Oenat_Mt__Extended____Nat__Oenat_J,type,
    compow4567540516116640754d_enat: nat > ( extended_enat > extended_enat ) > extended_enat > extended_enat ).

thf(sy_c_Nat_Ocompow_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    compow_int_int: nat > ( int > int ) > int > int ).

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_Ocompow_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    compow_real_real: nat > ( real > real ) > real > real ).

thf(sy_c_Nat_Ofunpow_001t__Nat__Onat,type,
    funpow_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_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

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__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Extended____Nat__Oenat,type,
    semiri8563196900006977889d_enat: ( extended_enat > extended_enat ) > nat > extended_enat > extended_enat ).

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__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_It__Int__Oint_J_J,type,
    size_s533118279054570080st_int: list_list_int > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__Int__Oint_Mt__Int__Oint_J_J,type,
    size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    size_s7647898544948552527nt_nat: list_P8198026277950538467nt_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__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_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

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

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

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__Real__Oreal,type,
    neg_nu6075765906172075777c_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__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__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_001t__Int__Oint,type,
    some_int: int > option_int ).

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_Osize__option_001t__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_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__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,
    bot_bo4898103413517107610_nat_o: product_prod_nat_nat > product_prod_nat_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__VEBT____Definitions__OVEBT_M_062_It__Extended____Nat__Oenat_M_Eo_J_J,type,
    bot_bo2578006069712851624enat_o: vEBT_VEBT > extended_enat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bo1565574316222977092_nat_o: vEBT_VEBT > 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__Filter__Ofilter_It__Nat__Onat_J_J,type,
    bot_bo498966703094740906er_nat: set_filter_nat ).

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__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__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__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J_J,type,
    bot_bo4330027929424010533d_enat: set_Pr2457182780427864761d_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    bot_bo1642239108664514429BT_nat: set_Pr7556676689462069481BT_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__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_Oless_001_Eo,type,
    ord_less_o: $o > $o > $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__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_less_filter_nat: filter_nat > filter_nat > $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__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__Filter__Ofilter_It__Nat__Onat_J_J,type,
    ord_le6505334834405097322er_nat: set_filter_nat > set_filter_nat > $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__Real__Oreal_J,type,
    ord_less_set_real: set_real > set_real > $o ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

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,
    ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Extended____Nat__Oenat_M_Eo_J_J,type,
    ord_le2691948842708570076enat_o: ( vEBT_VEBT > extended_enat > $o ) > ( vEBT_VEBT > extended_enat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__VEBT____Definitions__OVEBT_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le1182472622972956176_nat_o: ( vEBT_VEBT > nat > $o ) > ( vEBT_VEBT > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
    ord_less_eq_o: $o > $o > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
    ord_le2426478655948331894er_nat: set_filter_nat > set_filter_nat > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J_J,type,
    ord_le8566326065971749465d_enat: set_Pr2457182780427864761d_enat > set_Pr2457182780427864761d_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    ord_le3442269383143156041BT_nat: set_Pr7556676689462069481BT_nat > set_Pr7556676689462069481BT_nat > $o ).

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_max_filter_nat: filter_nat > filter_nat > filter_nat ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    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__Int__Oint,type,
    ord_min_int: int > int > int ).

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

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

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

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

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

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
    product_Pair_int_o: int > $o > product_prod_int_o ).

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

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
    product_Pair_int_nat: int > nat > product_prod_int_nat ).

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

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).

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

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

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

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

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

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

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

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

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_I_Eo_J,type,
    produc4257766111578684402_set_o: ( int > int > set_o ) > product_prod_int_int > set_o ).

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

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

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
    produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).

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

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
    produc59986286002894506_set_o: ( nat > nat > set_o ) > product_prod_nat_nat > set_o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    produc6189476227299908564et_nat: ( nat > nat > set_nat ) > product_prod_nat_nat > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc6744312701629110395at_nat: ( product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ) > produc859450856879609959at_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    produc4174022389229927035d_enat: ( vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ) > produc7272778201969148633d_enat > produc7272778201969148633d_enat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6408287024330202629at_nat: produc859450856879609959at_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
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thf(sy_c_Product__Type_Ounit_ORep__unit,type,
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thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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thf(sy_c_Relation_OField_001t__Nat__Onat,type,
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thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
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thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osums_001t__Int__Oint,type,
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thf(sy_c_Series_Osums_001t__Nat__Onat,type,
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thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Set_OCollect_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,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,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set_Ois__singleton_001_Eo,type,
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thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
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thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
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thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Othe__elem_001_Eo,type,
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thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
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thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
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thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
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thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Extended____Nat__Oenat,type,
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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__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
    set_or5403411693681687835d_enat: extended_enat > extended_enat > set_Extended_enat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    set_or1955772592623580779er_nat: filter_nat > filter_nat > set_filter_nat ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_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_001_Eo,type,
    set_or7139685690850216873Than_o: $o > $o > set_o ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Extended____Nat__Oenat,type,
    set_or4374356025156299511d_enat: extended_enat > extended_enat > set_Extended_enat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    set_or1773934645810362255er_nat: filter_nat > filter_nat > set_filter_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_OatLeastLessThan_001t__Real__Oreal,type,
    set_or66887138388493659n_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).

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

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

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Extended____Nat__Oenat,type,
    set_or8332593352340944941d_enat: extended_enat > set_Extended_enat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    set_or9144418160755794905er_nat: filter_nat > set_filter_nat ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_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__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_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_Omonoseq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    topolo6427056007704750605er_nat: ( nat > filter_nat ) > $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__Real__Oreal,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: ( nat > set_nat ) > $o ).

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

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

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: ( nat > complex ) > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: complex > complex ).

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

thf(sy_c_Transcendental_Osinh_001t__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__Complex__Ocomplex,type,
    tan_complex: complex > complex ).

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

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

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

thf(sy_c_Transfer_Obi__total_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bi_tot896582865486249351at_int: ( product_prod_nat_nat > int > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__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_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_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

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

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

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

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

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

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
    wf_int: set_Pr958786334691620121nt_int > $o ).

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

thf(sy_c_member_001_Eo,type,
    member_o2: $o > set_o > $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__Filter__Ofilter_It__Nat__Onat_J,type,
    member_filter_nat: filter_nat > set_filter_nat > $o ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int2: int > set_int > $o ).

thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: list_int > set_list_int > $o ).

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

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

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat2: nat > set_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    member4489920277610959864_int_o: product_prod_int_o > set_Pr903927857289325719_int_o > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
    member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    member8549952807677709168T_VEBT: produc8025551001238799321T_VEBT > set_Pr6167073792073659919T_VEBT > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
    member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    member38198578724832770d_enat: produc7272778201969148633d_enat > set_Pr2457182780427864761d_enat > $o ).

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real2: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat2: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT2: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_i____,type,
    i: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9086)
thf(fact_0_True,axiom,
    ( i
    = ( vEBT_VEBT_high @ xa @ na ) ) ).

% True
thf(fact_1__C166_C,axiom,
    vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ).

% "166"
thf(fact_2_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_3__C164_C,axiom,
    ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ i )
    = ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).

% "164"
thf(fact_4__C162_C,axiom,
    ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ X_1 ) ).

% "162"
thf(fact_5_insprop,axiom,
    ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) )
    = ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).

% insprop
thf(fact_6_tc,axiom,
    vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) ).

% tc
thf(fact_7_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N ) ) @ ( vEBT_VEBT_low @ X2 @ N ) ) ) ) ).

% in_children_def
thf(fact_8__C161_C,axiom,
    ~ ? [X_1: nat] : ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ X_1 ) ).

% "161"
thf(fact_9__C165_C,axiom,
    vEBT_invar_vebt @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) @ na ).

% "165"
thf(fact_10_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_11_list__update__id,axiom,
    ! [Xs: list_int,I: nat] :
      ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_12_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_13_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_14_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
        = ( nth_int @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_15_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_16__C163_C,axiom,
    ~ ( vEBT_V8194947554948674370ptions @ summary @ i ) ).

% "163"
thf(fact_17_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_18_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_19_nsprop,axiom,
    ( ~ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) )
   => ( summary
      = ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).

% nsprop
thf(fact_20__C11_C,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ) )
     => ( vEBT_invar_vebt @ X3 @ na ) ) ).

% "11"
thf(fact_21_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X4: vEBT_VEBT] :
      ( ( I != I2 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I2 @ X4 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X4 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_22_False,axiom,
    ~ ( ( xa = mi )
      | ( xa = ma ) ) ).

% False
thf(fact_23_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_24_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_25_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_26__C5_C,axiom,
    ( ( mi = ma )
   => ! [X3: vEBT_VEBT] :
        ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ).

% "5"
thf(fact_27_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat2 @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_28__C0_C,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ X3 @ na ) ) ).

% "0"
thf(fact_29_mimaxrel,axiom,
    ( ( xa != mi )
    & ( xa != ma ) ) ).

% mimaxrel
thf(fact_30__C1_C,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "1"
thf(fact_31_abcdef,axiom,
    ord_less_nat @ mi @ xa ).

% abcdef
thf(fact_32__C5_Ohyps_C_I7_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5.hyps"(7)
thf(fact_33__C12_C,axiom,
    vEBT_invar_vebt @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) @ summary ) @ m ).

% "12"
thf(fact_34_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_35_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_36_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_37_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_38_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_39_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_40__C10_C,axiom,
    ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( ord_max_nat @ xa @ ma ) ) ) @ deg @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) @ summary ) ) ) ).

% "10"
thf(fact_41_in__set__member,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
      = ( member_real @ Xs @ X ) ) ).

% in_set_member
thf(fact_42_in__set__member,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
      = ( member_o @ Xs @ X ) ) ).

% in_set_member
thf(fact_43_in__set__member,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
      = ( member_set_nat @ Xs @ X ) ) ).

% in_set_member
thf(fact_44_in__set__member,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( member_VEBT_VEBT @ Xs @ X ) ) ).

% in_set_member
thf(fact_45_in__set__member,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( member_int @ Xs @ X ) ) ).

% in_set_member
thf(fact_46_in__set__member,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( member_nat @ Xs @ X ) ) ).

% in_set_member
thf(fact_47_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( Deg = N2 ) ) ).

% deg_deg_n
thf(fact_48_in__set__insert,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ( ( insert_real @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_49_in__set__insert,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ( ( insert_o @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_50_in__set__insert,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( insert_set_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_51_in__set__insert,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( insert_VEBT_VEBT @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_52_in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( insert_int @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_53_in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_54_list__ex1__iff,axiom,
    ( list_ex1_real
    = ( ^ [P: real > $o,Xs2: list_real] :
        ? [X2: real] :
          ( ( member_real2 @ X2 @ ( set_real2 @ Xs2 ) )
          & ( P @ X2 )
          & ! [Y2: real] :
              ( ( ( member_real2 @ Y2 @ ( set_real2 @ Xs2 ) )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_55_list__ex1__iff,axiom,
    ( list_ex1_o
    = ( ^ [P: $o > $o,Xs2: list_o] :
        ? [X2: $o] :
          ( ( member_o2 @ X2 @ ( set_o2 @ Xs2 ) )
          & ( P @ X2 )
          & ! [Y2: $o] :
              ( ( ( member_o2 @ Y2 @ ( set_o2 @ Xs2 ) )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_56_list__ex1__iff,axiom,
    ( list_ex1_set_nat
    = ( ^ [P: set_nat > $o,Xs2: list_set_nat] :
        ? [X2: set_nat] :
          ( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs2 ) )
          & ( P @ X2 )
          & ! [Y2: set_nat] :
              ( ( ( member_set_nat2 @ Y2 @ ( set_set_nat2 @ Xs2 ) )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_57_list__ex1__iff,axiom,
    ( list_ex1_VEBT_VEBT
    = ( ^ [P: vEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
        ? [X2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
          & ( P @ X2 )
          & ! [Y2: vEBT_VEBT] :
              ( ( ( member_VEBT_VEBT2 @ Y2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_58_list__ex1__iff,axiom,
    ( list_ex1_int
    = ( ^ [P: int > $o,Xs2: list_int] :
        ? [X2: int] :
          ( ( member_int2 @ X2 @ ( set_int2 @ Xs2 ) )
          & ( P @ X2 )
          & ! [Y2: int] :
              ( ( ( member_int2 @ Y2 @ ( set_int2 @ Xs2 ) )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_59_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P: nat > $o,Xs2: list_nat] :
        ? [X2: nat] :
          ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X2 )
          & ! [Y2: nat] :
              ( ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs2 ) )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_60_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_61_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 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_62__C3_C,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "3"
thf(fact_63_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).

% buildup_gives_valid
thf(fact_64_mem__Collect__eq,axiom,
    ! [A: $o,P2: $o > $o] :
      ( ( member_o2 @ A @ ( collect_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
    ! [A: real,P2: real > $o] :
      ( ( member_real2 @ A @ ( collect_real @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( member_set_nat2 @ A @ ( collect_set_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
    ! [A: int,P2: int > $o] :
      ( ( member_int2 @ A @ ( collect_int @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A2: set_o] :
      ( ( collect_o
        @ ^ [X2: $o] : ( member_o2 @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X2: real] : ( member_real2 @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int2 @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_76_Collect__cong,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P2 @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_real @ P2 )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_77_Collect__cong,axiom,
    ! [P2: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P2 @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_list_nat @ P2 )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_78_Collect__cong,axiom,
    ! [P2: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P2 @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_set_nat @ P2 )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_79_Collect__cong,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P2 @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_80_Collect__cong,axiom,
    ! [P2: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P2 @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_int @ P2 )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_81_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_82_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_83_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_84_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_85_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_86_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_87_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_88_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X @ Z )
        & ( ord_le72135733267957522d_enat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_89_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_90_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_91_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_92_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_93_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_94_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_95_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_96_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_97_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_98_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_99_max_Obounded__iff,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
      = ( ( ord_le2932123472753598470d_enat @ B @ A )
        & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_100_max_Obounded__iff,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
      = ( ( ord_less_eq_real @ B @ A )
        & ( ord_less_eq_real @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_101_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_102_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_103_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_104_max_Oabsorb2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_105_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_106_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_107_max_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ A )
      = A ) ).

% max.idem
thf(fact_108_max_Oidem,axiom,
    ! [A: int] :
      ( ( ord_max_int @ A @ A )
      = A ) ).

% max.idem
thf(fact_109_max_Oidem,axiom,
    ! [A: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ A @ A )
      = A ) ).

% max.idem
thf(fact_110_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
      = ( ord_max_nat @ A @ B ) ) ).

% max.left_idem
thf(fact_111_max_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
      = ( ord_max_int @ A @ B ) ) ).

% max.left_idem
thf(fact_112_max_Oleft__idem,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) )
      = ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.left_idem
thf(fact_113_max_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
      = ( ord_max_nat @ A @ B ) ) ).

% max.right_idem
thf(fact_114_max_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
      = ( ord_max_int @ A @ B ) ) ).

% max.right_idem
thf(fact_115_max_Oright__idem,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ ( ord_ma741700101516333627d_enat @ A @ B ) @ B )
      = ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.right_idem
thf(fact_116_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_117_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_118_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_119_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_120_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_121_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_122_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_123_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_124_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_125_max_Oabsorb1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_126_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_127_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_128_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_129_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_130_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_131_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_132_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_133_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_134_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_135_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_136_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_137_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_138_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_139_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_140_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_141_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_142_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_143_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_144_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_145_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_146_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_147_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_148_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_149_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_150_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N: nat] :
        ? [K2: nat] :
          ( N
          = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_151_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_152_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_153_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
      = ( ! [X2: real] :
            ( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
           => ( member_real2 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_154_subset__code_I1_J,axiom,
    ! [Xs: list_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
      = ( ! [X2: $o] :
            ( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
           => ( member_o2 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_155_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat2 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_156_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT2 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_157_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
      = ( ! [X2: int] :
            ( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
           => ( member_int2 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_158_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
           => ( member_nat2 @ X2 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_159_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_160_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_161_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
     => ( ( member_real2 @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_162_set__update__subsetI,axiom,
    ! [Xs: list_o,A2: set_o,X: $o,I: nat] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
     => ( ( member_o2 @ X @ A2 )
       => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_163_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
     => ( ( member_set_nat2 @ X @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_164_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
     => ( ( member_int2 @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_165_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
     => ( ( member_VEBT_VEBT2 @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_166_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat2 @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_167_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_168_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_169_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_170_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_171_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_172_nat__less__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
             => ( P2 @ M4 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N2 ) ) ).

% nat_less_induct
thf(fact_173_infinite__descent,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P2 @ N3 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
              & ~ ( P2 @ M4 ) ) )
     => ( P2 @ N2 ) ) ).

% infinite_descent
thf(fact_174_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_175_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_176_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_177_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_178_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_179_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_180_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X5: nat] :
            ( ( P2 @ X5 )
            & ! [Y4: nat] :
                ( ( P2 @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_181_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.assoc
thf(fact_182_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.assoc
thf(fact_183_max_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ ( ord_ma741700101516333627d_enat @ A @ B ) @ C )
      = ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ B @ C ) ) ) ).

% max.assoc
thf(fact_184_max_Ocommute,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_max_nat @ B3 @ A3 ) ) ) ).

% max.commute
thf(fact_185_max_Ocommute,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B3: int] : ( ord_max_int @ B3 @ A3 ) ) ) ).

% max.commute
thf(fact_186_max_Ocommute,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( ord_ma741700101516333627d_enat @ B3 @ A3 ) ) ) ).

% max.commute
thf(fact_187_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_188_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.left_commute
thf(fact_189_max_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ C ) )
      = ( ord_ma741700101516333627d_enat @ A @ ( ord_ma741700101516333627d_enat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_190_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_191_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_192_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_193_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_194_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_195_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_196_infinite__descent0,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P2 @ N3 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N3 )
                  & ~ ( P2 @ M4 ) ) ) )
       => ( P2 @ N2 ) ) ) ).

% infinite_descent0
thf(fact_197_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_198_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_199_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_200_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_201_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N: nat] :
          ( ( ord_less_eq_nat @ M2 @ N )
          & ( M2 != N ) ) ) ) ).

% nat_less_le
thf(fact_202_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_203_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N: nat] :
          ( ( ord_less_nat @ M2 @ N )
          | ( M2 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_204_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_205_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_206_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_207_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ( ord_le2932123472753598470d_enat @ D @ B )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_208_max_Omono,axiom,
    ! [C: real,A: real,D: real,B: real] :
      ( ( ord_less_eq_real @ C @ A )
     => ( ( ord_less_eq_real @ D @ B )
       => ( ord_less_eq_real @ ( ord_max_real @ C @ D ) @ ( ord_max_real @ A @ B ) ) ) ) ).

% max.mono
thf(fact_209_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_210_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_211_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_212_max_OorderE,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( A
        = ( ord_max_real @ A @ B ) ) ) ).

% max.orderE
thf(fact_213_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_214_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_215_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_216_max_OorderI,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( ord_max_real @ A @ B ) )
     => ( ord_less_eq_real @ B @ A ) ) ).

% max.orderI
thf(fact_217_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_218_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_219_max_OboundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_220_max_OboundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_real @ B @ A )
         => ~ ( ord_less_eq_real @ C @ A ) ) ) ).

% max.boundedE
thf(fact_221_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_222_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_223_max_OboundedI,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ A )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_224_max_OboundedI,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ A )
       => ( ord_less_eq_real @ ( ord_max_real @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_225_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_226_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_227_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( A3
          = ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_228_max_Oorder__iff,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( A3
          = ( ord_max_real @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_229_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( A3
          = ( ord_max_nat @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_230_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( A3
          = ( ord_max_int @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_231_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_232_max_Ocobounded1,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ A @ ( ord_max_real @ A @ B ) ) ).

% max.cobounded1
thf(fact_233_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_234_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_235_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_236_max_Ocobounded2,axiom,
    ! [B: real,A: real] : ( ord_less_eq_real @ B @ ( ord_max_real @ A @ B ) ) ).

% max.cobounded2
thf(fact_237_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_238_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_239_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X )
        | ( ord_le2932123472753598470d_enat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_240_le__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_eq_real @ Z @ X )
        | ( ord_less_eq_real @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_241_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_242_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_243_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_244_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_max_real @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_245_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_max_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_246_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_max_int @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_247_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_248_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_max_real @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_249_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_max_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_250_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_max_int @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_251_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_252_max_OcoboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ A )
     => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_253_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_254_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_255_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_256_max_OcoboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ C @ B )
     => ( ord_less_eq_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_257_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_258_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_259_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_260_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X )
        | ( ord_le72135733267957522d_enat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_261_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_262_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_263_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_264_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_265_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_266_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_267_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( ord_max_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_268_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( A3
            = ( ord_ma741700101516333627d_enat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_269_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( A3
            = ( ord_max_real @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_270_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( A3
            = ( ord_max_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_271_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_272_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_273_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_274_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_275_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_276_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_277_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_278_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_279_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ N2 )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_280_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_281_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_282_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_283_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_284_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_285_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_286_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_287_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_288_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_289_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_290_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_291_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_292_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_293_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_294_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_295_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_296_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_297_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_298_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_299_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_300_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_301_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_302_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_303_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_304_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_305_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_306_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_307_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_308_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_309_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_310_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_311_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_312_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_313_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_314_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_315_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_316_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_317_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_318_le__zero__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% le_zero_eq
thf(fact_319_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_320_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_321_not__gr__zero,axiom,
    ! [N2: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% not_gr_zero
thf(fact_322_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_323_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_324_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_325_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_326_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_327_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_328_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_329_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_330_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_331_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_332_add__0,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
      = A ) ).

% add_0
thf(fact_333_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_334_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( zero_z5237406670263579293d_enat
        = ( plus_p3455044024723400733d_enat @ X @ Y ) )
      = ( ( X = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_335_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_336_add__eq__0__iff__both__eq__0,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
        = zero_z5237406670263579293d_enat )
      = ( ( X = zero_z5237406670263579293d_enat )
        & ( Y = zero_z5237406670263579293d_enat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_337_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_338_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_339_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_340_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_341_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_342_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_343_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_344_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ B @ A ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_345_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_346_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_347_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_348_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_349_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_350_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_351_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_352_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_353_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_354_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_355_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_356_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_357_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_358_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_359_add_Oright__neutral,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
      = A ) ).

% add.right_neutral
thf(fact_360_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_361_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_362_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_363_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_364_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_365_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_366_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_367_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_368_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_369_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_370_zero__reorient,axiom,
    ! [X: extended_enat] :
      ( ( zero_z5237406670263579293d_enat = X )
      = ( X = zero_z5237406670263579293d_enat ) ) ).

% zero_reorient
thf(fact_371_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_372_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_373_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_374_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_375_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_376_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_377_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_378_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_379_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_380_add_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) )
      = ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_381_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_382_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_383_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_384_add_Ocommute,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( plus_p3455044024723400733d_enat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_385_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_386_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_387_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_388_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_389_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_390_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_391_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_392_add_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
      = ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).

% add.assoc
thf(fact_393_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_394_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_395_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_396_group__cancel_Oadd2,axiom,
    ! [B2: extended_enat,K: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( B2
        = ( plus_p3455044024723400733d_enat @ K @ B ) )
     => ( ( plus_p3455044024723400733d_enat @ A @ B2 )
        = ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_397_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_398_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_399_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_400_group__cancel_Oadd1,axiom,
    ! [A2: extended_enat,K: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( A2
        = ( plus_p3455044024723400733d_enat @ K @ A ) )
     => ( ( plus_p3455044024723400733d_enat @ A2 @ B )
        = ( plus_p3455044024723400733d_enat @ K @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_401_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_402_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_403_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_404_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_p3455044024723400733d_enat @ I @ K )
        = ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_405_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_406_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_407_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_408_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
      = ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_409_zero__le,axiom,
    ! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).

% zero_le
thf(fact_410_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_411_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_412_zero__less__iff__neq__zero,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% zero_less_iff_neq_zero
thf(fact_413_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_414_gr__implies__not__zero,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M @ N2 )
     => ( N2 != zero_z5237406670263579293d_enat ) ) ).

% gr_implies_not_zero
thf(fact_415_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_416_not__less__zero,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_less_zero
thf(fact_417_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_418_gr__zeroI,axiom,
    ! [N2: extended_enat] :
      ( ( N2 != zero_z5237406670263579293d_enat )
     => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ).

% gr_zeroI
thf(fact_419_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_420_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_421_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_422_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_423_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_424_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_425_le__iff__add,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
        ? [C2: extended_enat] :
          ( B3
          = ( plus_p3455044024723400733d_enat @ A3 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_426_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C2: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_427_add__right__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_428_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_429_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_430_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_431_less__eqE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ~ ! [C3: extended_enat] :
            ( B
           != ( plus_p3455044024723400733d_enat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_432_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_433_add__left__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ C @ A ) @ ( plus_p3455044024723400733d_enat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_434_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_435_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_436_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_437_add__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ C @ D )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_438_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_439_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_440_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_441_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ I @ J )
        & ( ord_le2932123472753598470d_enat @ K @ L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_442_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_443_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_444_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_445_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( I = J )
        & ( ord_le2932123472753598470d_enat @ K @ L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_446_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_447_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_448_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_449_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
      ( ( ( ord_le2932123472753598470d_enat @ I @ J )
        & ( K = L ) )
     => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_450_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_451_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_452_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_453_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_454_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_455_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_456_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_457_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_458_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_459_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_460_add_Ocomm__neutral,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ zero_z5237406670263579293d_enat )
      = A ) ).

% add.comm_neutral
thf(fact_461_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_462_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_463_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_464_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_465_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ zero_z5237406670263579293d_enat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_466_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_467_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_468_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_469_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_470_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_471_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_472_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_473_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_474_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_475_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_476_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_477_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_478_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_479_add__strict__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ C @ D )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_480_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_481_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_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_491_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_492_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_493_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_494_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_495_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_496_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_497_add__nonpos__eq__0__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ Y @ zero_z5237406670263579293d_enat )
       => ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
            = zero_z5237406670263579293d_enat )
          = ( ( X = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_498_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_499_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_500_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_501_add__nonneg__eq__0__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ Y )
       => ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
            = zero_z5237406670263579293d_enat )
          = ( ( X = zero_z5237406670263579293d_enat )
            & ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_502_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_503_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_504_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_505_add__nonpos__nonpos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_nonpos
thf(fact_506_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_507_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_508_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_509_add__nonneg__nonneg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_510_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_511_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_512_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_513_add__increasing2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
     => ( ( ord_le2932123472753598470d_enat @ B @ A )
       => ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_514_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_515_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_516_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_517_add__decreasing2,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ A @ B )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_518_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_519_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_520_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_521_add__increasing,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_522_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_523_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_524_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_525_add__decreasing,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ C @ B )
       => ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_526_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_527_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_528_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_529_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_530_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_531_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_532_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_533_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_534_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_535_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_536_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_537_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_538_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_539_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_540_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_541_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_542_pos__add__strict,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_543_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_544_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_545_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_546_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ! [C3: extended_enat] :
            ( ( B
              = ( plus_p3455044024723400733d_enat @ A @ C3 ) )
           => ( C3 = zero_z5237406670263579293d_enat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_547_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_548_add__pos__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_549_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_550_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_551_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_552_add__neg__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_neg
thf(fact_553_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_554_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_555_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_556_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_557_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_558_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_559_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_560_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_561_add__pos__nonneg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_562_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_563_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_564_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_565_add__nonpos__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_neg
thf(fact_566_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_567_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_568_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_569_add__nonneg__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_570_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_571_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_572_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_573_add__neg__nonpos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_nonpos
thf(fact_574_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_575_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_576_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_577_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_578_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_579_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_580_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_581_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_582_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_583_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_584__C8_C,axiom,
    ( ( suc @ na )
    = m ) ).

% "8"
thf(fact_585_can__select__set__list__ex1,axiom,
    ! [P2: vEBT_VEBT > $o,A2: list_VEBT_VEBT] :
      ( ( can_select_VEBT_VEBT @ P2 @ ( set_VEBT_VEBT2 @ A2 ) )
      = ( list_ex1_VEBT_VEBT @ P2 @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_586_can__select__set__list__ex1,axiom,
    ! [P2: int > $o,A2: list_int] :
      ( ( can_select_int @ P2 @ ( set_int2 @ A2 ) )
      = ( list_ex1_int @ P2 @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_587_can__select__set__list__ex1,axiom,
    ! [P2: nat > $o,A2: list_nat] :
      ( ( can_select_nat @ P2 @ ( set_nat2 @ A2 ) )
      = ( list_ex1_nat @ P2 @ A2 ) ) ).

% can_select_set_list_ex1
thf(fact_588_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_589_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_590_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_591_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] :
          ( X
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_592_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_593_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_594_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_595_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_596_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_597_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_598_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_599_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_600_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_601_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_602_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_603_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_604_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_605_can__select__def,axiom,
    ( can_select_real
    = ( ^ [P: real > $o,A4: set_real] :
        ? [X2: real] :
          ( ( member_real2 @ X2 @ A4 )
          & ( P @ X2 )
          & ! [Y2: real] :
              ( ( ( member_real2 @ Y2 @ A4 )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_606_can__select__def,axiom,
    ( can_select_o
    = ( ^ [P: $o > $o,A4: set_o] :
        ? [X2: $o] :
          ( ( member_o2 @ X2 @ A4 )
          & ( P @ X2 )
          & ! [Y2: $o] :
              ( ( ( member_o2 @ Y2 @ A4 )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_607_can__select__def,axiom,
    ( can_select_set_nat
    = ( ^ [P: set_nat > $o,A4: set_set_nat] :
        ? [X2: set_nat] :
          ( ( member_set_nat2 @ X2 @ A4 )
          & ( P @ X2 )
          & ! [Y2: set_nat] :
              ( ( ( member_set_nat2 @ Y2 @ A4 )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_608_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P: nat > $o,A4: set_nat] :
        ? [X2: nat] :
          ( ( member_nat2 @ X2 @ A4 )
          & ( P @ X2 )
          & ! [Y2: nat] :
              ( ( ( member_nat2 @ Y2 @ A4 )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_609_can__select__def,axiom,
    ( can_select_int
    = ( ^ [P: int > $o,A4: set_int] :
        ? [X2: int] :
          ( ( member_int2 @ X2 @ A4 )
          & ( P @ X2 )
          & ! [Y2: int] :
              ( ( ( member_int2 @ Y2 @ A4 )
                & ( P @ Y2 ) )
             => ( Y2 = X2 ) ) ) ) ) ).

% can_select_def
thf(fact_610_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_611_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_612_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_613_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_614_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_615_nat__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P2 @ N3 )
           => ( P2 @ ( suc @ N3 ) ) )
       => ( P2 @ N2 ) ) ) ).

% nat_induct
thf(fact_616_diff__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X5: nat] : ( P2 @ X5 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X5: nat,Y3: nat] :
              ( ( P2 @ X5 @ Y3 )
             => ( P2 @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
         => ( P2 @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_617_zero__induct,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ! [N3: nat] :
            ( ( P2 @ ( suc @ N3 ) )
           => ( P2 @ N3 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_618_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_619_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_620_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_621_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_622_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_623_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_624_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_625_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_626_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_627_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_628_Ex__less__Suc,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
            & ( P2 @ I5 ) ) )
      = ( ( P2 @ N2 )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ N2 )
            & ( P2 @ I5 ) ) ) ) ).

% Ex_less_Suc
thf(fact_629_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_630_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_631_All__less__Suc,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
           => ( P2 @ I5 ) ) )
      = ( ( P2 @ N2 )
        & ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ N2 )
           => ( P2 @ I5 ) ) ) ) ).

% All_less_Suc
thf(fact_632_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N2 @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_633_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_634_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_635_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_636_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P2 @ I3 @ J2 )
                 => ( ( P2 @ J2 @ K3 )
                   => ( P2 @ I3 @ K3 ) ) ) ) )
         => ( P2 @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_637_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P2 @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P2 @ ( suc @ I3 ) )
               => ( P2 @ I3 ) ) )
         => ( P2 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_638_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_639_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_640_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_641_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_642_Suc__le__D,axiom,
    ! [N2: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
     => ? [M3: nat] :
          ( M6
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_643_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_644_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_645_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_646_full__nat__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
             => ( P2 @ M4 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N2 ) ) ).

% full_nat_induct
thf(fact_647_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P2 @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P2 @ N3 )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_648_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_649_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_650_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_651_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_652_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_653_lift__Suc__mono__less,axiom,
    ! [F: nat > extended_enat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_le72135733267957522d_enat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_654_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_655_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_656_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_657_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > extended_enat,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_le72135733267957522d_enat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_658_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_659_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_660_lift__Suc__mono__le,axiom,
    ! [F: nat > filter_nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le2510731241096832064er_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_le2510731241096832064er_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_661_lift__Suc__mono__le,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_662_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_663_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_664_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_665_lift__Suc__antimono__le,axiom,
    ! [F: nat > filter_nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_le2510731241096832064er_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_le2510731241096832064er_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_666_lift__Suc__antimono__le,axiom,
    ! [F: nat > real,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_667_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_668_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_669_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_670_Ex__less__Suc2,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
            & ( P2 @ I5 ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ N2 )
            & ( P2 @ ( suc @ I5 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_671_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M2: nat] :
            ( N2
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_672_All__less__Suc2,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N2 ) )
           => ( P2 @ I5 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ N2 )
           => ( P2 @ ( suc @ I5 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_673_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_674_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_675_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_676_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_677_dec__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P2 @ N3 )
                 => ( P2 @ ( suc @ N3 ) ) ) ) )
         => ( P2 @ J ) ) ) ) ).

% dec_induct
thf(fact_678_inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P2 @ ( suc @ N3 ) )
                 => ( P2 @ N3 ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% inc_induct
thf(fact_679_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_680_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_681_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_682_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_683_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_684_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_685_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_686_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_687_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N: nat] :
        ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_688_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_689_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_690_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q3: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_691_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_692_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ N2 )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K3 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_693_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X3 ) ).

% linordered_field_no_lb
thf(fact_694_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_695_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_696_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_697_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_698_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_699_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_700_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_701_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_702_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_703_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_704_max__bot,axiom,
    ! [X: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X )
      = X ) ).

% max_bot
thf(fact_705_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_706_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_707_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_708_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_709_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_710_max__bot2,axiom,
    ! [X: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ X @ bot_bo4199563552545308370d_enat )
      = X ) ).

% max_bot2
thf(fact_711_subset__empty,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_712_subset__empty,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_713_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_714_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_715_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_716_empty__subsetI,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% empty_subsetI
thf(fact_717_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_718_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_719_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X2: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
          | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).

% both_member_options_def
thf(fact_720_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_721_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_722_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_723_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_724_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( member_real2 @ X5 @ B2 ) )
     => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% subsetI
thf(fact_725_subsetI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( member_o2 @ X5 @ B2 ) )
     => ( ord_less_eq_set_o @ A2 @ B2 ) ) ).

% subsetI
thf(fact_726_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat2 @ X5 @ A2 )
         => ( member_set_nat2 @ X5 @ B2 ) )
     => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_727_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( member_int2 @ X5 @ B2 ) )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% subsetI
thf(fact_728_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( member_nat2 @ X5 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_729_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% psubsetI
thf(fact_730_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_731_dual__order_Orefl,axiom,
    ! [A: filter_nat] : ( ord_le2510731241096832064er_nat @ A @ A ) ).

% dual_order.refl
thf(fact_732_dual__order_Orefl,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% dual_order.refl
thf(fact_733_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_734_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_735_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_736_order__refl,axiom,
    ! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).

% order_refl
thf(fact_737_order__refl,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ X ) ).

% order_refl
thf(fact_738_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_739_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_740_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_741_empty__Collect__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P2 ) )
      = ( ! [X2: list_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_742_empty__Collect__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P2 ) )
      = ( ! [X2: set_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_743_empty__Collect__eq,axiom,
    ! [P2: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P2 ) )
      = ( ! [X2: real] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_744_empty__Collect__eq,axiom,
    ! [P2: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P2 ) )
      = ( ! [X2: $o] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_745_empty__Collect__eq,axiom,
    ! [P2: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P2 ) )
      = ( ! [X2: nat] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_746_empty__Collect__eq,axiom,
    ! [P2: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P2 ) )
      = ( ! [X2: int] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_747_Collect__empty__eq,axiom,
    ! [P2: list_nat > $o] :
      ( ( ( collect_list_nat @ P2 )
        = bot_bot_set_list_nat )
      = ( ! [X2: list_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_748_Collect__empty__eq,axiom,
    ! [P2: set_nat > $o] :
      ( ( ( collect_set_nat @ P2 )
        = bot_bot_set_set_nat )
      = ( ! [X2: set_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_749_Collect__empty__eq,axiom,
    ! [P2: real > $o] :
      ( ( ( collect_real @ P2 )
        = bot_bot_set_real )
      = ( ! [X2: real] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_750_Collect__empty__eq,axiom,
    ! [P2: $o > $o] :
      ( ( ( collect_o @ P2 )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_751_Collect__empty__eq,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_752_Collect__empty__eq,axiom,
    ! [P2: int > $o] :
      ( ( ( collect_int @ P2 )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_753_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ! [X2: set_nat] :
            ~ ( member_set_nat2 @ X2 @ A2 ) )
      = ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_754_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ! [X2: real] :
            ~ ( member_real2 @ X2 @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_755_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ! [X2: $o] :
            ~ ( member_o2 @ X2 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_756_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat2 @ X2 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_757_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ! [X2: int] :
            ~ ( member_int2 @ X2 @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_758_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat2 @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_759_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real2 @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_760_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o2 @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_761_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat2 @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_762_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int2 @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_763_subset__antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% subset_antisym
thf(fact_764_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_765_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_766_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_767_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_768_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_769_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_770_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_771_order__antisym__conv,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( ord_le2510731241096832064er_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_772_order__antisym__conv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_773_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_774_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_775_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_776_linorder__le__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_777_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_778_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_779_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_780_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_781_ord__le__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_782_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_783_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_784_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_785_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_786_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_787_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_788_ord__le__eq__subst,axiom,
    ! [A: filter_nat,B: filter_nat,F: filter_nat > real,C: real] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: filter_nat,Y3: filter_nat] :
              ( ( ord_le2510731241096832064er_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_789_ord__eq__le__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_790_ord__eq__le__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_791_ord__eq__le__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_792_ord__eq__le__subst,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_793_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_794_ord__eq__le__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_795_ord__eq__le__subst,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_796_ord__eq__le__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_797_ord__eq__le__subst,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_798_ord__eq__le__subst,axiom,
    ! [A: real,F: filter_nat > real,B: filter_nat,C: filter_nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le2510731241096832064er_nat @ B @ C )
       => ( ! [X5: filter_nat,Y3: filter_nat] :
              ( ( ord_le2510731241096832064er_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_799_linorder__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_linear
thf(fact_800_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_801_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_802_order__eq__refl,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
     => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_803_order__eq__refl,axiom,
    ! [X: real,Y: real] :
      ( ( X = Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_eq_refl
thf(fact_804_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_805_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_806_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_807_order__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_808_order__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_809_order__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_810_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_811_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_812_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_813_order__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_814_order__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_815_order__subst2,axiom,
    ! [A: int,B: int,F: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_816_order__subst2,axiom,
    ! [A: filter_nat,B: filter_nat,F: filter_nat > real,C: real] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: filter_nat,Y3: filter_nat] :
              ( ( ord_le2510731241096832064er_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_817_order__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_818_order__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_819_order__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_820_order__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_821_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_822_order__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_823_order__subst1,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_824_order__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_825_order__subst1,axiom,
    ! [A: int,F: int > int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_826_order__subst1,axiom,
    ! [A: filter_nat,F: real > filter_nat,B: real,C: real] :
      ( ( ord_le2510731241096832064er_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_le2510731241096832064er_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le2510731241096832064er_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_827_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: filter_nat,Z3: filter_nat] : Y5 = Z3 )
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
          & ( ord_le2510731241096832064er_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_828_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: real,Z3: real] : Y5 = Z3 )
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_829_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_830_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_831_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_832_antisym,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( ord_le2510731241096832064er_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_833_antisym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_834_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_835_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_836_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_837_dual__order_Otrans,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B @ A )
     => ( ( ord_le2510731241096832064er_nat @ C @ B )
       => ( ord_le2510731241096832064er_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_838_dual__order_Otrans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_839_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_840_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_841_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_842_dual__order_Oantisym,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B @ A )
     => ( ( ord_le2510731241096832064er_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_843_dual__order_Oantisym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_844_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_845_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_846_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_847_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: filter_nat,Z3: filter_nat] : Y5 = Z3 )
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
          & ( ord_le2510731241096832064er_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_848_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: real,Z3: real] : Y5 = Z3 )
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_849_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_850_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_851_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_852_linorder__wlog,axiom,
    ! [P2: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B4: real] :
          ( ( ord_less_eq_real @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: real,B4: real] :
            ( ( P2 @ B4 @ A5 )
           => ( P2 @ A5 @ B4 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_853_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: nat,B4: nat] :
            ( ( P2 @ B4 @ A5 )
           => ( P2 @ A5 @ B4 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_854_linorder__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: int,B4: int] :
            ( ( P2 @ B4 @ A5 )
           => ( P2 @ A5 @ B4 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_855_order__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
       => ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_856_order__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_eq_real @ X @ Z ) ) ) ).

% order_trans
thf(fact_857_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_858_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_859_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_860_order_Otrans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( ord_le2510731241096832064er_nat @ B @ C )
       => ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_861_order_Otrans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% order.trans
thf(fact_862_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_863_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_864_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_865_order__antisym,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_866_order__antisym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_867_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_868_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_869_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_870_ord__le__eq__trans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( B = C )
       => ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_871_ord__le__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_872_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_873_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_874_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_875_ord__eq__le__trans,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( A = B )
     => ( ( ord_le2510731241096832064er_nat @ B @ C )
       => ( ord_le2510731241096832064er_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_876_ord__eq__le__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_877_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_878_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_879_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_880_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: filter_nat,Z3: filter_nat] : Y5 = Z3 )
    = ( ^ [X2: filter_nat,Y2: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
          & ( ord_le2510731241096832064er_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_881_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: real,Z3: real] : Y5 = Z3 )
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_eq_real @ X2 @ Y2 )
          & ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_882_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_883_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_884_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_885_le__cases3,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ( ord_less_eq_real @ X @ Y )
       => ~ ( ord_less_eq_real @ Y @ Z ) )
     => ( ( ( ord_less_eq_real @ Y @ X )
         => ~ ( ord_less_eq_real @ X @ Z ) )
       => ( ( ( ord_less_eq_real @ X @ Z )
           => ~ ( ord_less_eq_real @ Z @ Y ) )
         => ( ( ( ord_less_eq_real @ Z @ Y )
             => ~ ( ord_less_eq_real @ Y @ X ) )
           => ( ( ( ord_less_eq_real @ Y @ Z )
               => ~ ( ord_less_eq_real @ Z @ X ) )
             => ~ ( ( ord_less_eq_real @ Z @ X )
                 => ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_886_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_887_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_888_nle__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_eq_real @ A @ B ) )
      = ( ( ord_less_eq_real @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_889_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_890_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_891_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_892_order__less__imp__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_893_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_894_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_895_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_896_order__less__imp__not__eq2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_897_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_898_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_899_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_900_order__less__imp__not__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_901_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_902_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_903_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_904_linorder__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
      | ( X = Y )
      | ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_905_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_906_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_907_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_908_order__less__imp__triv,axiom,
    ! [X: extended_enat,Y: extended_enat,P2: $o] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_909_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P2: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_910_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_911_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_912_order__less__not__sym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_913_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_914_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_915_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_916_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_917_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_918_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_919_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_920_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_921_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_922_order__less__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_923_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_924_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_925_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_926_order__less__subst1,axiom,
    ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_927_order__less__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_928_order__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_929_order__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_930_order__less__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_931_order__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_932_order__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_933_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_934_order__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_935_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_936_order__less__irrefl,axiom,
    ! [X: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ X @ X ) ).

% order_less_irrefl
thf(fact_937_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_938_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_939_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_940_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_941_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_942_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_943_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_944_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_945_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_946_ord__less__eq__subst,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_947_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_948_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_949_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_950_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_951_ord__eq__less__subst,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_952_ord__eq__less__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_953_ord__eq__less__subst,axiom,
    ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_954_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_955_ord__eq__less__subst,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_956_ord__eq__less__subst,axiom,
    ! [A: int,F: extended_enat > int,B: extended_enat,C: extended_enat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_957_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_958_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_959_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_960_order__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z )
       => ( ord_le72135733267957522d_enat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_961_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_962_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_963_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_964_order__less__asym_H,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).

% order_less_asym'
thf(fact_965_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_966_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_967_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_968_linorder__neq__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      = ( ( ord_le72135733267957522d_enat @ X @ Y )
        | ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_969_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_970_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_971_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_972_order__less__asym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_asym
thf(fact_973_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_974_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_975_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_976_linorder__neqE,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
     => ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
       => ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_977_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_978_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_979_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_980_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_981_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_982_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_983_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_984_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_985_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_986_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_987_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_988_dual__order_Ostrict__trans,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_989_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_990_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_991_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_992_not__less__iff__gr__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_993_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_994_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_995_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_996_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_997_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_998_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_999_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: nat] : ( P2 @ A5 @ A5 )
       => ( ! [A5: nat,B4: nat] :
              ( ( P2 @ B4 @ A5 )
             => ( P2 @ A5 @ B4 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1000_linorder__less__wlog,axiom,
    ! [P2: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
      ( ! [A5: extended_enat,B4: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: extended_enat] : ( P2 @ A5 @ A5 )
       => ( ! [A5: extended_enat,B4: extended_enat] :
              ( ( P2 @ B4 @ A5 )
             => ( P2 @ A5 @ B4 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1001_linorder__less__wlog,axiom,
    ! [P2: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B4: real] :
          ( ( ord_less_real @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: real] : ( P2 @ A5 @ A5 )
       => ( ! [A5: real,B4: real] :
              ( ( P2 @ B4 @ A5 )
             => ( P2 @ A5 @ B4 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1002_linorder__less__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B4: int] :
          ( ( ord_less_int @ A5 @ B4 )
         => ( P2 @ A5 @ B4 ) )
     => ( ! [A5: int] : ( P2 @ A5 @ A5 )
       => ( ! [A5: int,B4: int] :
              ( ( P2 @ B4 @ A5 )
             => ( P2 @ A5 @ B4 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1003_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P: nat > $o] :
        ? [N: nat] :
          ( ( P @ N )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
             => ~ ( P @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1004_exists__least__iff,axiom,
    ( ( ^ [P3: extended_enat > $o] :
        ? [X6: extended_enat] : ( P3 @ X6 ) )
    = ( ^ [P: extended_enat > $o] :
        ? [N: extended_enat] :
          ( ( P @ N )
          & ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N )
             => ~ ( P @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1005_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1006_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1007_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_1008_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_1009_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1010_dual__order_Oasym,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1011_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_1012_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_1013_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1014_linorder__cases,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( X != Y )
       => ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1015_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1016_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1017_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1018_antisym__conv3,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ Y @ X )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1019_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1020_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1021_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X5 )
             => ( P2 @ Y4 ) )
         => ( P2 @ X5 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_1022_less__induct,axiom,
    ! [P2: extended_enat > $o,A: extended_enat] :
      ( ! [X5: extended_enat] :
          ( ! [Y4: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ Y4 @ X5 )
             => ( P2 @ Y4 ) )
         => ( P2 @ X5 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_1023_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1024_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( B = C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1025_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1026_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1027_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1028_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A = B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1029_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1030_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1031_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1032_order_Oasym,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).

% order.asym
thf(fact_1033_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_1034_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_1035_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1036_less__imp__neq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1037_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1038_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1039_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z2: real] :
          ( ( ord_less_real @ X @ Z2 )
          & ( ord_less_real @ Z2 @ Y ) ) ) ).

% dense
thf(fact_1040_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_1041_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).

% gt_ex
thf(fact_1042_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_1043_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_1044_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_1045_not__psubset__empty,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_1046_not__psubset__empty,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1047_not__psubset__empty,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1048_not__psubset__empty,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1049_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ? [X2: set_nat] : ( member_set_nat2 @ X2 @ A2 ) )
      = ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_1050_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ? [X2: real] : ( member_real2 @ X2 @ A2 ) )
      = ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_1051_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ? [X2: $o] : ( member_o2 @ X2 @ A2 ) )
      = ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_1052_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X2: nat] : ( member_nat2 @ X2 @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1053_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ? [X2: int] : ( member_int2 @ X2 @ A2 ) )
      = ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_1054_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat2 @ Y3 @ A2 )
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_1055_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real2 @ Y3 @ A2 )
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_1056_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o2 @ Y3 @ A2 )
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_1057_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat2 @ Y3 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1058_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int2 @ Y3 @ A2 )
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_1059_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ ( member_set_nat2 @ A @ A2 ) ) ).

% equals0D
thf(fact_1060_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ ( member_real2 @ A @ A2 ) ) ).

% equals0D
thf(fact_1061_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ ( member_o2 @ A @ A2 ) ) ).

% equals0D
thf(fact_1062_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat2 @ A @ A2 ) ) ).

% equals0D
thf(fact_1063_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ ( member_int2 @ A @ A2 ) ) ).

% equals0D
thf(fact_1064_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat2 @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_1065_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real2 @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_1066_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o2 @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_1067_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat2 @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_1068_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int2 @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_1069_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( ( ord_less_set_nat @ A4 @ B5 )
          | ( A4 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1070_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C4 )
       => ( ord_less_set_nat @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1071_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B5 )
          & ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1072_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ord_less_set_nat @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1073_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% psubset_imp_subset
thf(fact_1074_Collect__mono__iff,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q ) )
      = ( ! [X2: real] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1075_Collect__mono__iff,axiom,
    ! [P2: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q ) )
      = ( ! [X2: list_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1076_Collect__mono__iff,axiom,
    ! [P2: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) )
      = ( ! [X2: set_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1077_Collect__mono__iff,axiom,
    ! [P2: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q ) )
      = ( ! [X2: int] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1078_Collect__mono__iff,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
      = ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1079_set__eq__subset,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B5 )
          & ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_1080_subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_1081_Collect__mono,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P2 @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P2 ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_1082_Collect__mono,axiom,
    ! [P2: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P2 @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1083_Collect__mono,axiom,
    ! [P2: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P2 @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1084_Collect__mono,axiom,
    ! [P2: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P2 @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P2 ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1085_Collect__mono,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P2 @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1086_subset__refl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% subset_refl
thf(fact_1087_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A4: set_real,B5: set_real] :
        ! [T2: real] :
          ( ( member_real2 @ T2 @ A4 )
         => ( member_real2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1088_subset__iff,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A4: set_o,B5: set_o] :
        ! [T2: $o] :
          ( ( member_o2 @ T2 @ A4 )
         => ( member_o2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1089_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B5: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat2 @ T2 @ A4 )
         => ( member_set_nat2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1090_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B5: set_int] :
        ! [T2: int] :
          ( ( member_int2 @ T2 @ A4 )
         => ( member_int2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1091_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat2 @ T2 @ A4 )
         => ( member_nat2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1092_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B5 )
          & ( A4 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_1093_equalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% equalityD2
thf(fact_1094_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% equalityD1
thf(fact_1095_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A4: set_real,B5: set_real] :
        ! [X2: real] :
          ( ( member_real2 @ X2 @ A4 )
         => ( member_real2 @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1096_subset__eq,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A4: set_o,B5: set_o] :
        ! [X2: $o] :
          ( ( member_o2 @ X2 @ A4 )
         => ( member_o2 @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1097_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B5: set_set_nat] :
        ! [X2: set_nat] :
          ( ( member_set_nat2 @ X2 @ A4 )
         => ( member_set_nat2 @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1098_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B5: set_int] :
        ! [X2: int] :
          ( ( member_int2 @ X2 @ A4 )
         => ( member_int2 @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1099_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
        ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A4 )
         => ( member_nat2 @ X2 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1100_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).

% equalityE
thf(fact_1101_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).

% psubsetE
thf(fact_1102_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real2 @ C @ A2 )
       => ( member_real2 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1103_subsetD,axiom,
    ! [A2: set_o,B2: set_o,C: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ( member_o2 @ C @ A2 )
       => ( member_o2 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1104_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat2 @ C @ A2 )
       => ( member_set_nat2 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1105_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int2 @ C @ A2 )
       => ( member_int2 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1106_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat2 @ C @ A2 )
       => ( member_nat2 @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1107_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real2 @ X @ A2 )
       => ( member_real2 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1108_in__mono,axiom,
    ! [A2: set_o,B2: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ( member_o2 @ X @ A2 )
       => ( member_o2 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1109_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat2 @ X @ A2 )
       => ( member_set_nat2 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1110_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int2 @ X @ A2 )
       => ( member_int2 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1111_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat2 @ X @ A2 )
       => ( member_nat2 @ X @ B2 ) ) ) ).

% in_mono
thf(fact_1112_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1113_order__le__imp__less__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1114_order__le__imp__less__or__eq,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_less_filter_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1115_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1116_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1117_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1118_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1119_linorder__le__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
      | ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1120_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1121_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1122_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_1123_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1124_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1125_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1126_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > extended_enat,C: extended_enat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1127_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1128_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1129_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1130_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1131_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1132_order__less__le__subst2,axiom,
    ! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1133_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1134_order__less__le__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1135_order__less__le__subst1,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1136_order__less__le__subst1,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1137_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1138_order__less__le__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1139_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1140_order__less__le__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1141_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1142_order__less__le__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1143_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1144_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1145_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1146_order__le__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_eq_real @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1147_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1148_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1149_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1150_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1151_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1152_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1153_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1154_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1155_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1156_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1157_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1158_order__le__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1159_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1160_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1161_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1162_order__le__less__subst1,axiom,
    ! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y3: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1163_order__less__le__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le2932123472753598470d_enat @ Y @ Z )
       => ( ord_le72135733267957522d_enat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1164_order__less__le__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_less_filter_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
       => ( ord_less_filter_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1165_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1166_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1167_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1168_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1169_order__le__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z )
       => ( ord_le72135733267957522d_enat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1170_order__le__less__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_less_filter_nat @ Y @ Z )
       => ( ord_less_filter_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1171_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1172_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1173_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1174_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1175_order__neq__le__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != B )
     => ( ( ord_le2932123472753598470d_enat @ A @ B )
       => ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1176_order__neq__le__trans,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( A != B )
     => ( ( ord_le2510731241096832064er_nat @ A @ B )
       => ( ord_less_filter_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1177_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1178_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1179_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1180_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1181_order__le__neq__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( A != B )
       => ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1182_order__le__neq__trans,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_filter_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1183_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1184_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1185_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1186_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1187_order__less__imp__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1188_order__less__imp__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_less_filter_nat @ X @ Y )
     => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1189_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1190_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1191_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1192_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1193_linorder__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
      = ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1194_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1195_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1196_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1197_linorder__not__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ X @ Y ) )
      = ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1198_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1199_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1200_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1201_order__less__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X2: extended_enat,Y2: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1202_order__less__le,axiom,
    ( ord_less_filter_nat
    = ( ^ [X2: filter_nat,Y2: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1203_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_eq_real @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1204_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1205_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1206_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1207_order__le__less,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [X2: extended_enat,Y2: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1208_order__le__less,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [X2: filter_nat,Y2: filter_nat] :
          ( ( ord_less_filter_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1209_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_real @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1210_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_set_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1211_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1212_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_int @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1213_dual__order_Ostrict__implies__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1214_dual__order_Ostrict__implies__order,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ( ord_less_filter_nat @ B @ A )
     => ( ord_le2510731241096832064er_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1215_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1216_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1217_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1218_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1219_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ord_le2932123472753598470d_enat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1220_order_Ostrict__implies__order,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( ord_less_filter_nat @ A @ B )
     => ( ord_le2510731241096832064er_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1221_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1222_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1223_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1224_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1225_dual__order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ~ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1226_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_filter_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
          & ~ ( ord_le2510731241096832064er_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1227_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1228_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1229_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1230_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1231_dual__order_Ostrict__trans2,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1232_dual__order_Ostrict__trans2,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( ( ord_less_filter_nat @ B @ A )
     => ( ( ord_le2510731241096832064er_nat @ C @ B )
       => ( ord_less_filter_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1233_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1234_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1235_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1236_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1237_dual__order_Ostrict__trans1,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1238_dual__order_Ostrict__trans1,axiom,
    ! [B: filter_nat,A: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B @ A )
     => ( ( ord_less_filter_nat @ C @ B )
       => ( ord_less_filter_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1239_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1240_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1241_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1242_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1243_dual__order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1244_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_filter_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1245_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1246_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1247_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1248_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1249_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1250_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [B3: filter_nat,A3: filter_nat] :
          ( ( ord_less_filter_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1251_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_real @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1252_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_set_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1253_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1254_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_int @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1255_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_1256_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_1257_order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ~ ( ord_le2932123472753598470d_enat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1258_order_Ostrict__iff__not,axiom,
    ( ord_less_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
          & ~ ( ord_le2510731241096832064er_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1259_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1260_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1261_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1262_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1263_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1264_order_Ostrict__trans2,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( ord_less_filter_nat @ A @ B )
     => ( ( ord_le2510731241096832064er_nat @ B @ C )
       => ( ord_less_filter_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1265_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1266_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1267_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1268_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1269_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1270_order_Ostrict__trans1,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ B )
     => ( ( ord_less_filter_nat @ B @ C )
       => ( ord_less_filter_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1271_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1272_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1273_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1274_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1275_order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1276_order_Ostrict__iff__order,axiom,
    ( ord_less_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1277_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1278_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1279_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1280_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1281_order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1282_order_Oorder__iff__strict,axiom,
    ( ord_le2510731241096832064er_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] :
          ( ( ord_less_filter_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1283_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1284_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1285_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1286_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1287_not__le__imp__less,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ord_le72135733267957522d_enat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1288_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1289_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1290_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1291_less__le__not__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X2: extended_enat,Y2: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
          & ~ ( ord_le2932123472753598470d_enat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1292_less__le__not__le,axiom,
    ( ord_less_filter_nat
    = ( ^ [X2: filter_nat,Y2: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X2 @ Y2 )
          & ~ ( ord_le2510731241096832064er_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1293_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_eq_real @ X2 @ Y2 )
          & ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1294_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ~ ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1295_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1296_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_1297_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y )
         => ( ord_less_eq_real @ X5 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_1298_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z @ X5 )
         => ( ord_less_eq_real @ Y @ X5 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_1299_antisym__conv2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1300_antisym__conv2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ~ ( ord_less_filter_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1301_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1302_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1303_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1304_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1305_antisym__conv1,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le2932123472753598470d_enat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1306_antisym__conv1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ~ ( ord_less_filter_nat @ X @ Y )
     => ( ( ord_le2510731241096832064er_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1307_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1308_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1309_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1310_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1311_nless__le,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1312_nless__le,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( ~ ( ord_less_filter_nat @ A @ B ) )
      = ( ~ ( ord_le2510731241096832064er_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1313_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1314_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1315_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1316_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1317_leI,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).

% leI
thf(fact_1318_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_1319_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_1320_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_1321_leD,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).

% leD
thf(fact_1322_leD,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ~ ( ord_less_filter_nat @ X @ Y ) ) ).

% leD
thf(fact_1323_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_1324_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_1325_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_1326_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_1327_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_1328_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_1329_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_1330_bot_Oextremum__uniqueI,axiom,
    ! [A: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ bot_bot_filter_nat )
     => ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1331_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1332_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1333_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_1334_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_1335_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_1336_bot_Oextremum__unique,axiom,
    ! [A: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ A @ bot_bot_filter_nat )
      = ( A = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
thf(fact_1337_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_1338_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1339_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_1340_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_1341_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1342_bot_Oextremum,axiom,
    ! [A: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A ) ).

% bot.extremum
thf(fact_1343_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1344_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1345_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1346_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1347_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1348_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1349_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1350_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
      = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1351_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_1352_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1353_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1354_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1355_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1356_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).

% bot.extremum_strict
thf(fact_1357_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va2: nat] :
              ( X
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_1358_max__absorb2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_ma741700101516333627d_enat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1359_max__absorb2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ X @ Y )
     => ( ( ord_max_filter_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1360_max__absorb2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_max_real @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1361_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_max_set_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1362_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1363_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_1364_max__absorb1,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ( ord_ma741700101516333627d_enat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1365_max__absorb1,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ Y @ X )
     => ( ( ord_max_filter_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1366_max__absorb1,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ( ord_max_real @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1367_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_max_set_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1368_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1369_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_1370_max__def,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1371_max__def,axiom,
    ( ord_max_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] : ( if_filter_nat @ ( ord_le2510731241096832064er_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1372_max__def,axiom,
    ( ord_max_real
    = ( ^ [A3: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1373_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1374_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1375_max__def,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def
thf(fact_1376_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1377_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X22: num] :
      ( ( size_option_num @ X @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1378_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A6: nat,B6: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1379_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A6: product_prod_nat_nat,B6: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1380_old_Oprod_Oinject,axiom,
    ! [A: vEBT_VEBT,B: nat,A6: vEBT_VEBT,B6: nat] :
      ( ( ( produc738532404422230701BT_nat @ A @ B )
        = ( produc738532404422230701BT_nat @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1381_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A6: int,B6: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1382_old_Oprod_Oinject,axiom,
    ! [A: vEBT_VEBT,B: extended_enat,A6: vEBT_VEBT,B6: extended_enat] :
      ( ( ( produc581526299967858633d_enat @ A @ B )
        = ( produc581526299967858633d_enat @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1383_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_1384_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_1385_prod_Oinject,axiom,
    ! [X1: vEBT_VEBT,X22: nat,Y1: vEBT_VEBT,Y22: nat] :
      ( ( ( produc738532404422230701BT_nat @ X1 @ X22 )
        = ( produc738532404422230701BT_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_1386_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( ( product_Pair_int_int @ X1 @ X22 )
        = ( product_Pair_int_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_1387_prod_Oinject,axiom,
    ! [X1: vEBT_VEBT,X22: extended_enat,Y1: vEBT_VEBT,Y22: extended_enat] :
      ( ( ( produc581526299967858633d_enat @ X1 @ X22 )
        = ( produc581526299967858633d_enat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_1388_inthall,axiom,
    ! [Xs: list_real,P2: real > $o,N2: nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ ( set_real2 @ Xs ) )
         => ( P2 @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
       => ( P2 @ ( nth_real @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_1389_inthall,axiom,
    ! [Xs: list_o,P2: $o > $o,N2: nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ ( set_o2 @ Xs ) )
         => ( P2 @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
       => ( P2 @ ( nth_o @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_1390_inthall,axiom,
    ! [Xs: list_set_nat,P2: set_nat > $o,N2: nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat2 @ X5 @ ( set_set_nat2 @ Xs ) )
         => ( P2 @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P2 @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_1391_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,N2: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P2 @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P2 @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_1392_inthall,axiom,
    ! [Xs: list_int,P2: int > $o,N2: nat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
         => ( P2 @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
       => ( P2 @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_1393_inthall,axiom,
    ! [Xs: list_nat,P2: nat > $o,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
         => ( P2 @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( P2 @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_1394_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_1395_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_1396_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1397_Euclid__induct,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2 @ A5 @ B4 )
          = ( P2 @ B4 @ A5 ) )
     => ( ! [A5: nat] : ( P2 @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B4: nat] :
              ( ( P2 @ A5 @ B4 )
             => ( P2 @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
         => ( P2 @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1398_nat__descend__induct,axiom,
    ! [N2: nat,P2: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N2 @ K3 )
         => ( P2 @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K3 @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K3 ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_1399_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X5: set_nat] :
          ~ ( member_set_nat2 @ X5 @ A2 )
     => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_1400_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X5: real] :
          ~ ( member_real2 @ X5 @ A2 )
     => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_1401_subset__emptyI,axiom,
    ! [A2: set_o] :
      ( ! [X5: $o] :
          ~ ( member_o2 @ X5 @ A2 )
     => ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_1402_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X5: int] :
          ~ ( member_int2 @ X5 @ A2 )
     => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_1403_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X5: nat] :
          ~ ( member_nat2 @ X5 @ A2 )
     => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1404_exists__least__lemma,axiom,
    ! [P2: nat > $o] :
      ( ~ ( P2 @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P2 @ X_1 )
       => ? [N3: nat] :
            ( ~ ( P2 @ N3 )
            & ( P2 @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1405_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_1406_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1407_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_1408_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_1409_List_Ofinite__set,axiom,
    ! [Xs: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1410_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_list_update
thf(fact_1411_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_list_update
thf(fact_1412_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_1413_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_1414_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
     => ( ( list_update_int @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_1415_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_1416_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1417_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1418_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1419_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_1420_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_1421_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_1422_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1423_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
       != ( size_size_list_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1424_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1425_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1426_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1427_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_1428_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1429_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1430_size__neq__size__imp__neq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_VEBT_VEBT @ X )
       != ( size_size_VEBT_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1431_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1432_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1433_length__induct,axiom,
    ! [P2: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_1434_length__induct,axiom,
    ! [P2: list_int > $o,Xs: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_1435_length__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_1436_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1437_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1438_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1439_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1440_finite__list,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ? [Xs3: list_Extended_enat] :
          ( ( set_Extended_enat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1441_infinite__growing,axiom,
    ! [X7: set_o] :
      ( ( X7 != bot_bot_set_o )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ X7 )
           => ? [Xa: $o] :
                ( ( member_o2 @ Xa @ X7 )
                & ( ord_less_o @ X5 @ Xa ) ) )
       => ~ ( finite_finite_o @ X7 ) ) ) ).

% infinite_growing
thf(fact_1442_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ X7 )
           => ? [Xa: nat] :
                ( ( member_nat2 @ Xa @ X7 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X7 ) ) ) ).

% infinite_growing
thf(fact_1443_infinite__growing,axiom,
    ! [X7: set_Extended_enat] :
      ( ( X7 != bot_bo7653980558646680370d_enat )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ X7 )
           => ? [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ X7 )
                & ( ord_le72135733267957522d_enat @ X5 @ Xa ) ) )
       => ~ ( finite4001608067531595151d_enat @ X7 ) ) ) ).

% infinite_growing
thf(fact_1444_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ X7 )
           => ? [Xa: real] :
                ( ( member_real2 @ Xa @ X7 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X7 ) ) ) ).

% infinite_growing
thf(fact_1445_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ X7 )
           => ? [Xa: int] :
                ( ( member_int2 @ Xa @ X7 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X7 ) ) ) ).

% infinite_growing
thf(fact_1446_ex__min__if__finite,axiom,
    ! [S3: set_o] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o2 @ X5 @ S3 )
            & ~ ? [Xa: $o] :
                  ( ( member_o2 @ Xa @ S3 )
                  & ( ord_less_o @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1447_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat2 @ X5 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat2 @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1448_ex__min__if__finite,axiom,
    ! [S3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ S3 )
            & ~ ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ S3 )
                  & ( ord_le72135733267957522d_enat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1449_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real2 @ X5 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real2 @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1450_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int2 @ X5 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int2 @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1451_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I3 )
              = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_1452_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I3 )
              = ( nth_int @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_1453_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_1454_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > vEBT_VEBT > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X8: vEBT_VEBT] : ( P2 @ I5 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P2 @ I5 @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1455_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > int > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X8: int] : ( P2 @ I5 @ X8 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P2 @ I5 @ ( nth_int @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1456_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X8: nat] : ( P2 @ I5 @ X8 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P2 @ I5 @ ( nth_nat @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1457_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : Y5 = Z3 )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                = ( nth_VEBT_VEBT @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1458_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z3: list_int] : Y5 = Z3 )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I5 )
                = ( nth_int @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1459_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z3: list_nat] : Y5 = Z3 )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I5 )
                = ( nth_nat @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1460_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1461_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1462_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1463_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1464_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1465_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1466_nth__mem,axiom,
    ! [N2: nat,Xs: list_real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
     => ( member_real2 @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1467_nth__mem,axiom,
    ! [N2: nat,Xs: list_o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( member_o2 @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1468_nth__mem,axiom,
    ! [N2: nat,Xs: list_set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat2 @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1469_nth__mem,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT2 @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1470_nth__mem,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( member_int2 @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1471_nth__mem,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1472_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P2 @ X5 ) )
       => ( P2 @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1473_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_int,P2: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
           => ( P2 @ X5 ) )
       => ( P2 @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1474_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_nat,P2: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X5 ) )
       => ( P2 @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1475_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1476_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1477_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1478_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1479_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1480_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1481_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P2: real > $o,X: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
         => ( P2 @ ( nth_real @ Xs @ I3 ) ) )
     => ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1482_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P2: $o > $o,X: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
         => ( P2 @ ( nth_o @ Xs @ I3 ) ) )
     => ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1483_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P2: set_nat > $o,X: set_nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P2 @ ( nth_set_nat @ Xs @ I3 ) ) )
     => ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1484_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) )
     => ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1485_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: int > $o,X: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
         => ( P2 @ ( nth_int @ Xs @ I3 ) ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1486_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: nat > $o,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I3 ) ) )
     => ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1487_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P2 @ X2 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P2 @ ( nth_VEBT_VEBT @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1488_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
           => ( P2 @ X2 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
           => ( P2 @ ( nth_int @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1489_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X2 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
           => ( P2 @ ( nth_nat @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1490_set__update__memI,axiom,
    ! [N2: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
     => ( member_real2 @ X @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1491_set__update__memI,axiom,
    ! [N2: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( member_o2 @ X @ ( set_o2 @ ( list_update_o @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1492_set__update__memI,axiom,
    ! [N2: nat,Xs: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat2 @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1493_set__update__memI,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1494_set__update__memI,axiom,
    ! [N2: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( member_int2 @ X @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1495_set__update__memI,axiom,
    ! [N2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1496_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1497_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1498_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1499_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1500_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1501_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1502_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B4: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1503_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1504_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9072475918466114483BT_nat] :
      ~ ! [A5: vEBT_VEBT,B4: nat] :
          ( Y
         != ( produc738532404422230701BT_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1505_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A5: int,B4: int] :
          ( Y
         != ( product_Pair_int_int @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1506_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7272778201969148633d_enat] :
      ~ ! [A5: vEBT_VEBT,B4: extended_enat] :
          ( Y
         != ( produc581526299967858633d_enat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1507_surj__pair,axiom,
    ! [P4: product_prod_nat_nat] :
    ? [X5: nat,Y3: nat] :
      ( P4
      = ( product_Pair_nat_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_1508_surj__pair,axiom,
    ! [P4: produc859450856879609959at_nat] :
    ? [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P4
      = ( produc6161850002892822231at_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_1509_surj__pair,axiom,
    ! [P4: produc9072475918466114483BT_nat] :
    ? [X5: vEBT_VEBT,Y3: nat] :
      ( P4
      = ( produc738532404422230701BT_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_1510_surj__pair,axiom,
    ! [P4: product_prod_int_int] :
    ? [X5: int,Y3: int] :
      ( P4
      = ( product_Pair_int_int @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_1511_surj__pair,axiom,
    ! [P4: produc7272778201969148633d_enat] :
    ? [X5: vEBT_VEBT,Y3: extended_enat] :
      ( P4
      = ( produc581526299967858633d_enat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_1512_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
      ( ! [A5: nat,B4: nat] : ( P2 @ ( product_Pair_nat_nat @ A5 @ B4 ) )
     => ( P2 @ P4 ) ) ).

% prod_cases
thf(fact_1513_prod__cases,axiom,
    ! [P2: produc859450856879609959at_nat > $o,P4: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] : ( P2 @ ( produc6161850002892822231at_nat @ A5 @ B4 ) )
     => ( P2 @ P4 ) ) ).

% prod_cases
thf(fact_1514_prod__cases,axiom,
    ! [P2: produc9072475918466114483BT_nat > $o,P4: produc9072475918466114483BT_nat] :
      ( ! [A5: vEBT_VEBT,B4: nat] : ( P2 @ ( produc738532404422230701BT_nat @ A5 @ B4 ) )
     => ( P2 @ P4 ) ) ).

% prod_cases
thf(fact_1515_prod__cases,axiom,
    ! [P2: product_prod_int_int > $o,P4: product_prod_int_int] :
      ( ! [A5: int,B4: int] : ( P2 @ ( product_Pair_int_int @ A5 @ B4 ) )
     => ( P2 @ P4 ) ) ).

% prod_cases
thf(fact_1516_prod__cases,axiom,
    ! [P2: produc7272778201969148633d_enat > $o,P4: produc7272778201969148633d_enat] :
      ( ! [A5: vEBT_VEBT,B4: extended_enat] : ( P2 @ ( produc581526299967858633d_enat @ A5 @ B4 ) )
     => ( P2 @ P4 ) ) ).

% prod_cases
thf(fact_1517_Pair__inject,axiom,
    ! [A: nat,B: nat,A6: nat,B6: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1518_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A6: product_prod_nat_nat,B6: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1519_Pair__inject,axiom,
    ! [A: vEBT_VEBT,B: nat,A6: vEBT_VEBT,B6: nat] :
      ( ( ( produc738532404422230701BT_nat @ A @ B )
        = ( produc738532404422230701BT_nat @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1520_Pair__inject,axiom,
    ! [A: int,B: int,A6: int,B6: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1521_Pair__inject,axiom,
    ! [A: vEBT_VEBT,B: extended_enat,A6: vEBT_VEBT,B6: extended_enat] :
      ( ( ( produc581526299967858633d_enat @ A @ B )
        = ( produc581526299967858633d_enat @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1522_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B4: nat,C3: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ C3 ) ) ) ).

% prod_cases3
thf(fact_1523_prod__induct3,axiom,
    ! [P2: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: nat,C3: nat] : ( P2 @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ C3 ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_1524_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1525_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1526_ssubst__Pair__rhs,axiom,
    ! [R2: vEBT_VEBT,S: nat,R: set_Pr7556676689462069481BT_nat,S4: nat] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1527_ssubst__Pair__rhs,axiom,
    ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1528_ssubst__Pair__rhs,axiom,
    ! [R2: vEBT_VEBT,S: extended_enat,R: set_Pr2457182780427864761d_enat,S4: extended_enat] :
      ( ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1529_finite__has__minimal,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1530_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o2 @ X5 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o2 @ Xa @ A2 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1531_finite__has__minimal,axiom,
    ! [A2: set_filter_nat] :
      ( ( finite2119507909894593271er_nat @ A2 )
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ? [X5: filter_nat] :
            ( ( member_filter_nat @ X5 @ A2 )
            & ! [Xa: filter_nat] :
                ( ( member_filter_nat @ Xa @ A2 )
               => ( ( ord_le2510731241096832064er_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1532_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real2 @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real2 @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1533_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat2 @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1534_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat2 @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1535_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int2 @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1536_finite__has__maximal,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1537_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o2 @ X5 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o2 @ Xa @ A2 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1538_finite__has__maximal,axiom,
    ! [A2: set_filter_nat] :
      ( ( finite2119507909894593271er_nat @ A2 )
     => ( ( A2 != bot_bo498966703094740906er_nat )
       => ? [X5: filter_nat] :
            ( ( member_filter_nat @ X5 @ A2 )
            & ! [Xa: filter_nat] :
                ( ( member_filter_nat @ Xa @ A2 )
               => ( ( ord_le2510731241096832064er_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1539_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real2 @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real2 @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1540_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat2 @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1541_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat2 @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1542_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int2 @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1543_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1544_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X2: real] : ( member_real2 @ X2 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_1545_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X2: $o] : ( member_o2 @ X2 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_1546_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X2: nat] : ( member_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1547_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X2: int] : ( member_int2 @ X2 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1548_Collect__empty__eq__bot,axiom,
    ! [P2: list_nat > $o] :
      ( ( ( collect_list_nat @ P2 )
        = bot_bot_set_list_nat )
      = ( P2 = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1549_Collect__empty__eq__bot,axiom,
    ! [P2: set_nat > $o] :
      ( ( ( collect_set_nat @ P2 )
        = bot_bot_set_set_nat )
      = ( P2 = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1550_Collect__empty__eq__bot,axiom,
    ! [P2: real > $o] :
      ( ( ( collect_real @ P2 )
        = bot_bot_set_real )
      = ( P2 = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1551_Collect__empty__eq__bot,axiom,
    ! [P2: $o > $o] :
      ( ( ( collect_o @ P2 )
        = bot_bot_set_o )
      = ( P2 = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1552_Collect__empty__eq__bot,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( P2 = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1553_Collect__empty__eq__bot,axiom,
    ! [P2: int > $o] :
      ( ( ( collect_int @ P2 )
        = bot_bot_set_int )
      = ( P2 = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1554_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) @ M )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N2 @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1555_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1556_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1557_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M2: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M2 @ N )
            & ( member_nat2 @ N @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1558_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M2: nat] :
        ! [X2: nat] :
          ( ( member_nat2 @ X2 @ N5 )
         => ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1559_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N6: nat] :
              ( ( ord_less_nat @ M3 @ N6 )
              & ( member_nat2 @ N6 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_1560_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ N7 )
         => ( ord_less_nat @ X5 @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1561_length__enumerate,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_enumerate
thf(fact_1562_length__enumerate,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_enumerate
thf(fact_1563_length__enumerate,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_1564_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N3: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X3 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X3 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1565_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N3: nat] :
        ! [X3: list_int] :
          ( ( member_list_int @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X3 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1566_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N3: nat] :
        ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1567_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1568_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1569_subrelI,axiom,
    ! [R2: set_Pr7556676689462069481BT_nat,S: set_Pr7556676689462069481BT_nat] :
      ( ! [X5: vEBT_VEBT,Y3: nat] :
          ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X5 @ Y3 ) @ R2 )
         => ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X5 @ Y3 ) @ S ) )
     => ( ord_le3442269383143156041BT_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1570_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y3 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y3 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_1571_subrelI,axiom,
    ! [R2: set_Pr2457182780427864761d_enat,S: set_Pr2457182780427864761d_enat] :
      ( ! [X5: vEBT_VEBT,Y3: extended_enat] :
          ( ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X5 @ Y3 ) @ R2 )
         => ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X5 @ Y3 ) @ S ) )
     => ( ord_le8566326065971749465d_enat @ R2 @ S ) ) ).

% subrelI
thf(fact_1572_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X: nat,M7: nat] :
      ( ( P2 @ X )
     => ( ! [X5: nat] :
            ( ( P2 @ X5 )
           => ( ord_less_eq_nat @ X5 @ M7 ) )
       => ~ ! [M3: nat] :
              ( ( P2 @ M3 )
             => ~ ! [X3: nat] :
                    ( ( P2 @ X3 )
                   => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1573_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A5: nat,B4: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_1574_finite__has__minimal2,axiom,
    ! [A2: set_o,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o2 @ A @ A2 )
       => ? [X5: $o] :
            ( ( member_o2 @ X5 @ A2 )
            & ( ord_less_eq_o @ X5 @ A )
            & ! [Xa: $o] :
                ( ( member_o2 @ Xa @ A2 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1575_finite__has__minimal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ( ord_le2932123472753598470d_enat @ X5 @ A )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1576_finite__has__minimal2,axiom,
    ! [A2: set_filter_nat,A: filter_nat] :
      ( ( finite2119507909894593271er_nat @ A2 )
     => ( ( member_filter_nat @ A @ A2 )
       => ? [X5: filter_nat] :
            ( ( member_filter_nat @ X5 @ A2 )
            & ( ord_le2510731241096832064er_nat @ X5 @ A )
            & ! [Xa: filter_nat] :
                ( ( member_filter_nat @ Xa @ A2 )
               => ( ( ord_le2510731241096832064er_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1577_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real2 @ A @ A2 )
       => ? [X5: real] :
            ( ( member_real2 @ X5 @ A2 )
            & ( ord_less_eq_real @ X5 @ A )
            & ! [Xa: real] :
                ( ( member_real2 @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1578_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat2 @ A @ A2 )
       => ? [X5: set_nat] :
            ( ( member_set_nat2 @ X5 @ A2 )
            & ( ord_less_eq_set_nat @ X5 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1579_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat2 @ A @ A2 )
       => ? [X5: nat] :
            ( ( member_nat2 @ X5 @ A2 )
            & ( ord_less_eq_nat @ X5 @ A )
            & ! [Xa: nat] :
                ( ( member_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1580_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int2 @ A @ A2 )
       => ? [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
            & ( ord_less_eq_int @ X5 @ A )
            & ! [Xa: int] :
                ( ( member_int2 @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1581_finite__has__maximal2,axiom,
    ! [A2: set_o,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o2 @ A @ A2 )
       => ? [X5: $o] :
            ( ( member_o2 @ X5 @ A2 )
            & ( ord_less_eq_o @ A @ X5 )
            & ! [Xa: $o] :
                ( ( member_o2 @ Xa @ A2 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1582_finite__has__maximal2,axiom,
    ! [A2: set_Extended_enat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ A @ A2 )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
            & ( ord_le2932123472753598470d_enat @ A @ X5 )
            & ! [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ A2 )
               => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1583_finite__has__maximal2,axiom,
    ! [A2: set_filter_nat,A: filter_nat] :
      ( ( finite2119507909894593271er_nat @ A2 )
     => ( ( member_filter_nat @ A @ A2 )
       => ? [X5: filter_nat] :
            ( ( member_filter_nat @ X5 @ A2 )
            & ( ord_le2510731241096832064er_nat @ A @ X5 )
            & ! [Xa: filter_nat] :
                ( ( member_filter_nat @ Xa @ A2 )
               => ( ( ord_le2510731241096832064er_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1584_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real2 @ A @ A2 )
       => ? [X5: real] :
            ( ( member_real2 @ X5 @ A2 )
            & ( ord_less_eq_real @ A @ X5 )
            & ! [Xa: real] :
                ( ( member_real2 @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1585_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat2 @ A @ A2 )
       => ? [X5: set_nat] :
            ( ( member_set_nat2 @ X5 @ A2 )
            & ( ord_less_eq_set_nat @ A @ X5 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1586_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat2 @ A @ A2 )
       => ? [X5: nat] :
            ( ( member_nat2 @ X5 @ A2 )
            & ( ord_less_eq_nat @ A @ X5 )
            & ! [Xa: nat] :
                ( ( member_nat2 @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1587_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int2 @ A @ A2 )
       => ? [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
            & ( ord_less_eq_int @ A @ X5 )
            & ! [Xa: int] :
                ( ( member_int2 @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1588_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [X5: set_nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: set_nat,Y3: set_nat,Z2: set_nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: set_nat] :
                ( ( member_set_nat2 @ X5 @ A2 )
               => ? [Y4: set_nat] :
                    ( ( member_set_nat2 @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1589_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: complex,Y3: complex,Z2: complex] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ A2 )
               => ? [Y4: complex] :
                    ( ( member_complex @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1590_finite__transitivity__chain,axiom,
    ! [A2: set_Extended_enat,R: extended_enat > extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: extended_enat,Y3: extended_enat,Z2: extended_enat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ A2 )
               => ? [Y4: extended_enat] :
                    ( ( member_Extended_enat @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1591_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: real,Y3: real,Z2: real] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ A2 )
               => ? [Y4: real] :
                    ( ( member_real2 @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1592_finite__transitivity__chain,axiom,
    ! [A2: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X5: $o] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: $o,Y3: $o,Z2: $o] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ A2 )
               => ? [Y4: $o] :
                    ( ( member_o2 @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1593_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat2 @ X5 @ A2 )
               => ? [Y4: nat] :
                    ( ( member_nat2 @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1594_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: int,Y3: int,Z2: int] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ A2 )
               => ? [Y4: int] :
                    ( ( member_int2 @ Y4 @ A2 )
                    & ( R @ X5 @ Y4 ) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1595_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_1596_infinite__imp__nonempty,axiom,
    ! [S3: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S3 )
     => ( S3 != bot_bo7653980558646680370d_enat ) ) ).

% infinite_imp_nonempty
thf(fact_1597_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ ( finite_finite_real @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_1598_infinite__imp__nonempty,axiom,
    ! [S3: set_o] :
      ( ~ ( finite_finite_o @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_1599_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1600_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ ( finite_finite_int @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_1601_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_1602_finite_OemptyI,axiom,
    finite4001608067531595151d_enat @ bot_bo7653980558646680370d_enat ).

% finite.emptyI
thf(fact_1603_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_1604_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_1605_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1606_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_1607_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1608_rev__finite__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1609_rev__finite__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1610_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1611_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S3 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_1612_infinite__super,axiom,
    ! [S3: set_int,T3: set_int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ T3 ) ) ) ).

% infinite_super
thf(fact_1613_infinite__super,axiom,
    ! [S3: set_Extended_enat,T3: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
     => ( ~ ( finite4001608067531595151d_enat @ S3 )
       => ~ ( finite4001608067531595151d_enat @ T3 ) ) ) ).

% infinite_super
thf(fact_1614_infinite__super,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_1615_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_1616_finite__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( finite_finite_int @ B2 )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_1617_finite__subset,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% finite_subset
thf(fact_1618_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_1619_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M2: nat] :
        ! [X2: nat] :
          ( ( member_nat2 @ X2 @ N5 )
         => ( ord_less_nat @ X2 @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1620_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M2: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M2 @ N )
            & ( member_nat2 @ N @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1621_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( ( member_complex @ X3 @ S3 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1622_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ S3 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic3845382081240766429at_nat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1623_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( ( member_real2 @ X3 @ S3 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic5055836439445974935al_nat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1624_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: $o > nat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X3: $o] :
              ( ( member_o2 @ X3 @ S3 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1625_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( ( member_nat2 @ X3 @ S3 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1626_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( ( member_int2 @ X3 @ S3 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1627_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( ( member_complex @ X3 @ S3 )
              & ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ ( lattic7796887085614042845d_enat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1628_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ~ ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ S3 )
              & ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ ( lattic1996716550891908761d_enat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1629_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( ( member_real2 @ X3 @ S3 )
              & ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ ( lattic9066027731366277983d_enat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1630_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_o,F: $o > extended_enat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ~ ? [X3: $o] :
              ( ( member_o2 @ X3 @ S3 )
              & ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ ( lattic7542187609180611815d_enat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1631_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_real @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1632_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S3 )
         => ( ord_less_eq_real @ ( F @ ( lattic1189837152898106425t_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1633_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real2 @ Y @ S3 )
         => ( ord_less_eq_real @ ( F @ ( lattic8440615504127631091l_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1634_arg__min__least,axiom,
    ! [S3: set_o,Y: $o,F: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ( ( member_o2 @ Y @ S3 )
         => ( ord_less_eq_real @ ( F @ ( lattic8697145971487455083o_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1635_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat2 @ Y @ S3 )
         => ( ord_less_eq_real @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1636_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int2 @ Y @ S3 )
         => ( ord_less_eq_real @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1637_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1638_arg__min__least,axiom,
    ! [S3: set_Extended_enat,Y: extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ( ( member_Extended_enat @ Y @ S3 )
         => ( ord_less_eq_nat @ ( F @ ( lattic3845382081240766429at_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1639_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real2 @ Y @ S3 )
         => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1640_arg__min__least,axiom,
    ! [S3: set_o,Y: $o,F: $o > nat] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ( ( member_o2 @ Y @ S3 )
         => ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S3 ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_1641_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_1642_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1643_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1644_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1645_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_1646_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1647_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_1648_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_1649_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_1650_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_1651_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1652_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).

% less_numeral_extra(3)
thf(fact_1653_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1654_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1655_complete__interval,axiom,
    ! [A: extended_enat,B: extended_enat,P2: extended_enat > $o] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C3: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ A @ C3 )
              & ( ord_le2932123472753598470d_enat @ C3 @ B )
              & ! [X3: extended_enat] :
                  ( ( ( ord_le2932123472753598470d_enat @ A @ X3 )
                    & ( ord_le72135733267957522d_enat @ X3 @ C3 ) )
                 => ( P2 @ X3 ) )
              & ! [D3: extended_enat] :
                  ( ! [X5: extended_enat] :
                      ( ( ( ord_le2932123472753598470d_enat @ A @ X5 )
                        & ( ord_le72135733267957522d_enat @ X5 @ D3 ) )
                     => ( P2 @ X5 ) )
                 => ( ord_le2932123472753598470d_enat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1656_complete__interval,axiom,
    ! [A: real,B: real,P2: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C3: real] :
              ( ( ord_less_eq_real @ A @ C3 )
              & ( ord_less_eq_real @ C3 @ B )
              & ! [X3: real] :
                  ( ( ( ord_less_eq_real @ A @ X3 )
                    & ( ord_less_real @ X3 @ C3 ) )
                 => ( P2 @ X3 ) )
              & ! [D3: real] :
                  ( ! [X5: real] :
                      ( ( ( ord_less_eq_real @ A @ X5 )
                        & ( ord_less_real @ X5 @ D3 ) )
                     => ( P2 @ X5 ) )
                 => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1657_complete__interval,axiom,
    ! [A: nat,B: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C3: nat] :
              ( ( ord_less_eq_nat @ A @ C3 )
              & ( ord_less_eq_nat @ C3 @ B )
              & ! [X3: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X3 )
                    & ( ord_less_nat @ X3 @ C3 ) )
                 => ( P2 @ X3 ) )
              & ! [D3: nat] :
                  ( ! [X5: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X5 )
                        & ( ord_less_nat @ X5 @ D3 ) )
                     => ( P2 @ X5 ) )
                 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1658_complete__interval,axiom,
    ! [A: int,B: int,P2: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C3: int] :
              ( ( ord_less_eq_int @ A @ C3 )
              & ( ord_less_eq_int @ C3 @ B )
              & ! [X3: int] :
                  ( ( ( ord_less_eq_int @ A @ X3 )
                    & ( ord_less_int @ X3 @ C3 ) )
                 => ( P2 @ X3 ) )
              & ! [D3: int] :
                  ( ! [X5: int] :
                      ( ( ( ord_less_eq_int @ A @ X5 )
                        & ( ord_less_int @ X5 @ D3 ) )
                     => ( P2 @ X5 ) )
                 => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1659_verit__comp__simplify1_I3_J,axiom,
    ! [B6: extended_enat,A6: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ B6 @ A6 ) )
      = ( ord_le72135733267957522d_enat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1660_verit__comp__simplify1_I3_J,axiom,
    ! [B6: real,A6: real] :
      ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
      = ( ord_less_real @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1661_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
      = ( ord_less_nat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1662_verit__comp__simplify1_I3_J,axiom,
    ! [B6: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
      = ( ord_less_int @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1663_pinf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ~ ( ord_le2932123472753598470d_enat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_1664_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ~ ( ord_less_eq_real @ X3 @ T ) ) ).

% pinf(6)
thf(fact_1665_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ~ ( ord_less_eq_nat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_1666_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ~ ( ord_less_eq_int @ X3 @ T ) ) ).

% pinf(6)
thf(fact_1667_verit__comp__simplify1_I2_J,axiom,
    ! [A: filter_nat] : ( ord_le2510731241096832064er_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1668_verit__comp__simplify1_I2_J,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1669_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1670_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1671_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1672_verit__la__disequality,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
      | ~ ( ord_less_eq_real @ A @ B )
      | ~ ( ord_less_eq_real @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1673_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1674_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1675_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ~ ( ord_less_nat @ T @ X3 ) ) ).

% minf(7)
thf(fact_1676_minf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ~ ( ord_le72135733267957522d_enat @ T @ X3 ) ) ).

% minf(7)
thf(fact_1677_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ~ ( ord_less_real @ T @ X3 ) ) ).

% minf(7)
thf(fact_1678_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ~ ( ord_less_int @ T @ X3 ) ) ).

% minf(7)
thf(fact_1679_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( ord_less_nat @ X3 @ T ) ) ).

% minf(5)
thf(fact_1680_minf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ( ord_le72135733267957522d_enat @ X3 @ T ) ) ).

% minf(5)
thf(fact_1681_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ( ord_less_real @ X3 @ T ) ) ).

% minf(5)
thf(fact_1682_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ( ord_less_int @ X3 @ T ) ) ).

% minf(5)
thf(fact_1683_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_1684_minf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_1685_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_1686_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_1687_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_1688_minf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_1689_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_1690_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_1691_minf_I2_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1692_minf_I2_J,axiom,
    ! [P2: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1693_minf_I2_J,axiom,
    ! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1694_minf_I2_J,axiom,
    ! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1695_minf_I1_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1696_minf_I1_J,axiom,
    ! [P2: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1697_minf_I1_J,axiom,
    ! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1698_minf_I1_J,axiom,
    ! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z4 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z4 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z2 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1699_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( ord_less_nat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_1700_pinf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ( ord_le72135733267957522d_enat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_1701_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( ord_less_real @ T @ X3 ) ) ).

% pinf(7)
thf(fact_1702_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ( ord_less_int @ T @ X3 ) ) ).

% pinf(7)
thf(fact_1703_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ~ ( ord_less_nat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_1704_pinf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ~ ( ord_le72135733267957522d_enat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_1705_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ~ ( ord_less_real @ X3 @ T ) ) ).

% pinf(5)
thf(fact_1706_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ~ ( ord_less_int @ X3 @ T ) ) ).

% pinf(5)
thf(fact_1707_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_1708_pinf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_1709_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_1710_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_1711_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_1712_pinf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_1713_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_1714_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_1715_pinf_I2_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1716_pinf_I2_J,axiom,
    ! [P2: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1717_pinf_I2_J,axiom,
    ! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1718_pinf_I2_J,axiom,
    ! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                | ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                | ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1719_pinf_I1_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q4: nat > $o] :
      ( ? [Z4: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1720_pinf_I1_J,axiom,
    ! [P2: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q4: extended_enat > $o] :
      ( ? [Z4: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1721_pinf_I1_J,axiom,
    ! [P2: real > $o,P5: real > $o,Q: real > $o,Q4: real > $o] :
      ( ? [Z4: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1722_pinf_I1_J,axiom,
    ! [P2: int > $o,P5: int > $o,Q: int > $o,Q4: int > $o] :
      ( ? [Z4: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z4 @ X5 )
         => ( ( P2 @ X5 )
            = ( P5 @ X5 ) ) )
     => ( ? [Z4: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z4 @ X5 )
           => ( ( Q @ X5 )
              = ( Q4 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z2 @ X3 )
           => ( ( ( P2 @ X3 )
                & ( Q @ X3 ) )
              = ( ( P5 @ X3 )
                & ( Q4 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1723_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1724_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1725_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1726_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1727_ex__gt__or__lt,axiom,
    ! [A: real] :
    ? [B4: real] :
      ( ( ord_less_real @ A @ B4 )
      | ( ord_less_real @ B4 @ A ) ) ).

% ex_gt_or_lt
thf(fact_1728_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1729_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1730_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K3: nat,M3: nat] :
          ( X
         != ( product_Pair_nat_nat @ K3 @ M3 ) ) ).

% prod_decode_aux.cases
thf(fact_1731_le__numeral__extra_I3_J,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).

% le_numeral_extra(3)
thf(fact_1732_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1733_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1734_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1735_minf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ~ ( ord_le2932123472753598470d_enat @ T @ X3 ) ) ).

% minf(8)
thf(fact_1736_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ~ ( ord_less_eq_real @ T @ X3 ) ) ).

% minf(8)
thf(fact_1737_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).

% minf(8)
thf(fact_1738_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ~ ( ord_less_eq_int @ T @ X3 ) ) ).

% minf(8)
thf(fact_1739_minf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ( ord_le2932123472753598470d_enat @ X3 @ T ) ) ).

% minf(6)
thf(fact_1740_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ( ord_less_eq_real @ X3 @ T ) ) ).

% minf(6)
thf(fact_1741_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( ord_less_eq_nat @ X3 @ T ) ) ).

% minf(6)
thf(fact_1742_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ( ord_less_eq_int @ X3 @ T ) ) ).

% minf(6)
thf(fact_1743_pinf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ( ord_le2932123472753598470d_enat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_1744_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( ord_less_eq_real @ T @ X3 ) ) ).

% pinf(8)
thf(fact_1745_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( ord_less_eq_nat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_1746_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ( ord_less_eq_int @ T @ X3 ) ) ).

% pinf(8)
thf(fact_1747_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_1748_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X )
        = ( find_P8199882355184865565at_nat @ P2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I5 ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1749_find__Some__iff2,axiom,
    ! [X: num,P2: num > $o,Xs: list_num] :
      ( ( ( some_num @ X )
        = ( find_num @ P2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_num @ Xs ) )
            & ( P2 @ ( nth_num @ Xs @ I5 ) )
            & ( X
              = ( nth_num @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1750_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( some_VEBT_VEBT @ X )
        = ( find_VEBT_VEBT @ P2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P2 @ ( nth_VEBT_VEBT @ Xs @ I5 ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1751_find__Some__iff2,axiom,
    ! [X: int,P2: int > $o,Xs: list_int] :
      ( ( ( some_int @ X )
        = ( find_int @ P2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
            & ( P2 @ ( nth_int @ Xs @ I5 ) )
            & ( X
              = ( nth_int @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1752_find__Some__iff2,axiom,
    ! [X: nat,P2: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X )
        = ( find_nat @ P2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
            & ( P2 @ ( nth_nat @ Xs @ I5 ) )
            & ( X
              = ( nth_nat @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1753_find__Some__iff,axiom,
    ! [P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P2 @ Xs )
        = ( some_P7363390416028606310at_nat @ X ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ I5 ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1754_find__Some__iff,axiom,
    ! [P2: num > $o,Xs: list_num,X: num] :
      ( ( ( find_num @ P2 @ Xs )
        = ( some_num @ X ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_num @ Xs ) )
            & ( P2 @ ( nth_num @ Xs @ I5 ) )
            & ( X
              = ( nth_num @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1755_find__Some__iff,axiom,
    ! [P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P2 @ Xs )
        = ( some_VEBT_VEBT @ X ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P2 @ ( nth_VEBT_VEBT @ Xs @ I5 ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1756_find__Some__iff,axiom,
    ! [P2: int > $o,Xs: list_int,X: int] :
      ( ( ( find_int @ P2 @ Xs )
        = ( some_int @ X ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
            & ( P2 @ ( nth_int @ Xs @ I5 ) )
            & ( X
              = ( nth_int @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1757_find__Some__iff,axiom,
    ! [P2: nat > $o,Xs: list_nat,X: nat] :
      ( ( ( find_nat @ P2 @ Xs )
        = ( some_nat @ X ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
            & ( P2 @ ( nth_nat @ Xs @ I5 ) )
            & ( X
              = ( nth_nat @ Xs @ I5 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I5 )
               => ~ ( P2 @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1758_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3941691890525107288d_enat @ Ys ) )
       => ( ( nth_Pr7509392720524132704d_enat @ ( zip_VE7205001627739651817d_enat @ Xs @ Ys ) @ I )
          = ( produc581526299967858633d_enat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_Extended_enat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1759_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I )
          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1760_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) @ I )
          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1761_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I )
          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1762_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) @ I )
          = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1763_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys ) @ I )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1764_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1765_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) @ I )
          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1766_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr3440142176431000676at_int @ ( zip_nat_int @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1767_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1768__C7_C,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [Y4: nat] :
              ( ( ( ( vEBT_VEBT_high @ Y4 @ na )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ Y4 @ na ) ) )
             => ( ( ord_less_nat @ mi @ Y4 )
                & ( ord_less_eq_nat @ Y4 @ ma ) ) ) ) ) ) ).

% "7"
thf(fact_1769_count__notin,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ( ( count_list_real @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1770_count__notin,axiom,
    ! [X: $o,Xs: list_o] :
      ( ~ ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ( ( count_list_o @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1771_count__notin,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( count_list_set_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1772_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( count_list_VEBT_VEBT @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1773_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( count_list_int @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1774_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( count_list_nat @ Xs @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1775_listrel1__iff__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
      = ( ? [Y2: product_prod_nat_nat,N: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ Y2 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( Ys
              = ( list_u6180841689913720943at_nat @ Xs @ N @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1776_listrel1__iff__update,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
      = ( ? [Y2: vEBT_VEBT,N: nat] :
            ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ Y2 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( Ys
              = ( list_u1324408373059187874T_VEBT @ Xs @ N @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1777_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
      = ( ? [Y2: int,N: nat] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ Y2 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
            & ( Ys
              = ( list_update_int @ Xs @ N @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1778_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y2: nat,N: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ Y2 ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1779_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1780_intind,axiom,
    ! [I: nat,N2: nat,P2: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P2 @ X )
       => ( P2 @ ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_1781_intind,axiom,
    ! [I: nat,N2: nat,P2: int > $o,X: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P2 @ X )
       => ( P2 @ ( nth_int @ ( replicate_int @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_1782_intind,axiom,
    ! [I: nat,N2: nat,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P2 @ X )
       => ( P2 @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_1783_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_1784__C2_C,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "2"
thf(fact_1785__C5_Oprems_C,axiom,
    ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.prems"
thf(fact_1786__C5_Ohyps_C_I8_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.hyps"(8)
thf(fact_1787__092_060open_062i_A_060_A2_A_094_Am_092_060close_062,axiom,
    ord_less_nat @ i @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).

% \<open>i < 2 ^ m\<close>
thf(fact_1788__C5_OIH_C_I1_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X3 @ na )
        & ! [Xa: nat] :
            ( ( ord_less_nat @ Xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
           => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ X3 @ Xa ) @ na ) ) ) ) ).

% "5.IH"(1)
thf(fact_1789__C4_C,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X8 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).

% "4"
thf(fact_1790_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_1791_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_1792_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1793_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1794_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1795_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1796_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1797_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1798_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1799_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1800_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_1801_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_1802_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_1803_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_1804_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1805_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1806_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1807_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1808_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_1809_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_1810_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_1811_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_1812_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X )
        = ( replicate_VEBT_VEBT @ N2 @ Y ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_1813_length__replicate,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_1814_length__replicate,axiom,
    ! [N2: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_1815_length__replicate,axiom,
    ! [N2: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X ) )
      = N2 ) ).

% length_replicate
thf(fact_1816_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_1817_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( 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_1818__C5_OIH_C_I2_J,axiom,
    ! [X: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ summary @ X ) @ m ) ) ).

% "5.IH"(2)
thf(fact_1819_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_1820_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_1821_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_1822_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_1823_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_1824_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_1825_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(3)
thf(fact_1826_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_1827_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_1828_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_1829_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(4)
thf(fact_1830_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_1831_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_1832_myIHs,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X @ na )
       => ( ( ord_less_nat @ Xa2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
         => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ X @ Xa2 ) @ na ) ) ) ) ).

% myIHs
thf(fact_1833__C6_C,axiom,
    ( ( ord_less_eq_nat @ mi @ ma )
    & ( ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ) ) ).

% "6"
thf(fact_1834_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P2: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int2 @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P2 @ X2 ) ) )
      = ( ( P2 @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1835_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P2: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
           => ( P2 @ X2 ) ) )
      = ( ( P2 @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1836_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P2 @ X2 ) ) )
      = ( ( P2 @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_1837_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P2: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int2 @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P2 @ X2 ) ) )
      = ( ( P2 @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1838_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P2: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
            & ( P2 @ X2 ) ) )
      = ( ( P2 @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1839_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ? [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P2 @ X2 ) ) )
      = ( ( P2 @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_1840_in__set__replicate,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( member_real2 @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1841_in__set__replicate,axiom,
    ! [X: $o,N2: nat,Y: $o] :
      ( ( member_o2 @ X @ ( set_o2 @ ( replicate_o @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1842_in__set__replicate,axiom,
    ! [X: set_nat,N2: nat,Y: set_nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1843_in__set__replicate,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( member_int2 @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1844_in__set__replicate,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1845_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
      = ( ( X = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1846_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_1847_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_1848_nth__replicate,axiom,
    ! [I: nat,N2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_1849_highlowprop,axiom,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% highlowprop
thf(fact_1850_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_1851_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_1852_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1853_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1854_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1855_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1856_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_1857_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1858_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1859_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_1860_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_1861_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1862_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_1863_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1864_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1865_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1866_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1867_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1868_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1869_listrel1__eq__len,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1870_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1871_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1872_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P2: real > $o,Q: real > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( set_real2 @ Ys ) )
           => ( ( P2 @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_real @ P2 @ Xs )
          = ( find_real @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1873_find__cong,axiom,
    ! [Xs: list_o,Ys: list_o,P2: $o > $o,Q: $o > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ ( set_o2 @ Ys ) )
           => ( ( P2 @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_o @ P2 @ Xs )
          = ( find_o @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1874_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: set_nat] :
            ( ( member_set_nat2 @ X5 @ ( set_set_nat2 @ Ys ) )
           => ( ( P2 @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_set_nat @ P2 @ Xs )
          = ( find_set_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1875_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P2: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( P2 @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_VEBT_VEBT @ P2 @ Xs )
          = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1876_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P2: int > $o,Q: int > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ ( set_int2 @ Ys ) )
           => ( ( P2 @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_int @ P2 @ Xs )
          = ( find_int @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1877_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P2: nat > $o,Q: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( set_nat2 @ Ys ) )
           => ( ( P2 @ X5 )
              = ( Q @ X5 ) ) )
       => ( ( find_nat @ P2 @ Xs )
          = ( find_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1878_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1879_set__zip__rightD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1880_set__zip__rightD,axiom,
    ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
     => ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1881_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1882_set__zip__rightD,axiom,
    ! [X: vEBT_VEBT,Y: extended_enat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X @ Y ) @ ( set_Pr4712219556205076590d_enat @ ( zip_VE7205001627739651817d_enat @ Xs @ Ys ) ) )
     => ( member_Extended_enat @ Y @ ( set_Extended_enat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1883_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1884_set__zip__leftD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) ) )
     => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1885_set__zip__leftD,axiom,
    ! [X: vEBT_VEBT,Y: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
     => ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1886_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1887_set__zip__leftD,axiom,
    ! [X: vEBT_VEBT,Y: extended_enat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X @ Y ) @ ( set_Pr4712219556205076590d_enat @ ( zip_VE7205001627739651817d_enat @ Xs @ Ys ) ) )
     => ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1888_in__set__zipE,axiom,
    ! [X: real,Y: real,Xs: list_real,Ys: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) )
     => ~ ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
         => ~ ( member_real2 @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1889_in__set__zipE,axiom,
    ! [X: real,Y: $o,Xs: list_real,Ys: list_o] :
      ( ( member772602641336174712real_o @ ( product_Pair_real_o @ X @ Y ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs @ Ys ) ) )
     => ~ ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
         => ~ ( member_o2 @ Y @ ( set_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1890_in__set__zipE,axiom,
    ! [X: $o,Y: real,Xs: list_o,Ys: list_real] :
      ( ( member7400031367953476362o_real @ ( product_Pair_o_real @ X @ Y ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs @ Ys ) ) )
     => ~ ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
         => ~ ( member_real2 @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1891_in__set__zipE,axiom,
    ! [X: $o,Y: $o,Xs: list_o,Ys: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) )
     => ~ ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
         => ~ ( member_o2 @ Y @ ( set_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1892_in__set__zipE,axiom,
    ! [X: real,Y: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
         => ~ ( member_VEBT_VEBT2 @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1893_in__set__zipE,axiom,
    ! [X: $o,Y: vEBT_VEBT,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X @ Y ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
         => ~ ( member_VEBT_VEBT2 @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1894_in__set__zipE,axiom,
    ! [X: real,Y: int,Xs: list_real,Ys: list_int] :
      ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) )
     => ~ ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
         => ~ ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1895_in__set__zipE,axiom,
    ! [X: $o,Y: int,Xs: list_o,Ys: list_int] :
      ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X @ Y ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) )
     => ~ ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
         => ~ ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1896_in__set__zipE,axiom,
    ! [X: real,Y: nat,Xs: list_real,Ys: list_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1897_in__set__zipE,axiom,
    ! [X: $o,Y: nat,Xs: list_o,Ys: list_nat] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X @ Y ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1898_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
      = ( ( member_real2 @ A @ ( set_real2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1899_zip__same,axiom,
    ! [A: $o,B: $o,Xs: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Xs ) ) )
      = ( ( member_o2 @ A @ ( set_o2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1900_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
      = ( ( member_set_nat2 @ A @ ( set_set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1901_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
      = ( ( member_VEBT_VEBT2 @ A @ ( set_VEBT_VEBT2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1902_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat2 @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1903_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
      = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1904_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
      = ( ( member_int2 @ A @ ( set_int2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1905_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ Y ) )
      = ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I @ ( produc537772716801021591T_VEBT @ X @ Y ) ) ) ).

% zip_update
thf(fact_1906_zip__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
      = ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1907_zip__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) @ ( list_u6180841689913720943at_nat @ Ys @ I @ Y ) )
      = ( list_u5003261594476800725at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) @ I @ ( produc6161850002892822231at_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1908_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( zip_VEBT_VEBT_nat @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
      = ( list_u2459188882655168453BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I @ ( produc738532404422230701BT_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1909_zip__update,axiom,
    ! [Xs: list_int,I: nat,X: int,Ys: list_int,Y: int] :
      ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X ) @ ( list_update_int @ Ys @ I @ Y ) )
      = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1910_zip__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Ys: list_Extended_enat,Y: extended_enat] :
      ( ( zip_VE7205001627739651817d_enat @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u3071683517702156500d_enat @ Ys @ I @ Y ) )
      = ( list_u2241885204319820103d_enat @ ( zip_VE7205001627739651817d_enat @ Xs @ Ys ) @ I @ ( produc581526299967858633d_enat @ X @ Y ) ) ) ).

% zip_update
thf(fact_1911_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_1912_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_1913_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_1914_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_1915_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1916_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_1917_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1918_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_1919_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1920_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_1921_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1922_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1923_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1924_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1925_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_1926_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_1927_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( X5 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1928_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
         => ( X5 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1929_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
         => ( X5 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1930_replicate__eqI,axiom,
    ! [Xs: list_real,N2: nat,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = N2 )
     => ( ! [Y3: real] :
            ( ( member_real2 @ Y3 @ ( set_real2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_real @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1931_replicate__eqI,axiom,
    ! [Xs: list_o,N2: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs )
        = N2 )
     => ( ! [Y3: $o] :
            ( ( member_o2 @ Y3 @ ( set_o2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_o @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1932_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N2: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = N2 )
     => ( ! [Y3: set_nat] :
            ( ( member_set_nat2 @ Y3 @ ( set_set_nat2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_set_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1933_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1934_replicate__eqI,axiom,
    ! [Xs: list_int,N2: nat,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = N2 )
     => ( ! [Y3: int] :
            ( ( member_int2 @ Y3 @ ( set_int2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_int @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1935_replicate__eqI,axiom,
    ! [Xs: list_nat,N2: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N2 )
     => ( ! [Y3: nat] :
            ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_nat @ N2 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_1936_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1937_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT,X: $o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X @ Y3 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1938_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ~ ! [Y3: int] :
              ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1939_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_int,X: $o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ~ ! [Y3: int] :
              ~ ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X @ Y3 ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1940_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1941_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_nat,X: $o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X @ Y3 ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1942_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_Extended_enat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s3941691890525107288d_enat @ Ys ) )
     => ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: extended_enat] :
              ~ ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X @ Y3 ) @ ( set_Pr4712219556205076590d_enat @ ( zip_VE7205001627739651817d_enat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1943_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1944_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: int] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1945_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1946_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real2 @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X5: vEBT_VEBT] :
              ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X5 @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1947_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Y: $o] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o2 @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X5: vEBT_VEBT] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X5 @ Y ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1948_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_Extended_enat,Y: extended_enat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s3941691890525107288d_enat @ Ys ) )
     => ( ( member_Extended_enat @ Y @ ( set_Extended_enat2 @ Ys ) )
       => ~ ! [X5: vEBT_VEBT] :
              ~ ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X5 @ Y ) @ ( set_Pr4712219556205076590d_enat @ ( zip_VE7205001627739651817d_enat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1949_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT2 @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X5: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X5 @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1950_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X5: vEBT_VEBT] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X5 @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1951_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat2 @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X5: vEBT_VEBT] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X5 @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1952_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_real,Y: real] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real2 @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X5: int] :
              ~ ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X5 @ Y ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1953_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_o,Y: $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o2 @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X5: int] :
              ~ ( member4489920277610959864_int_o @ ( product_Pair_int_o @ X5 @ Y ) @ ( set_Pr8694291782656941196_int_o @ ( zip_int_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1954_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT2 @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X5: int] :
              ~ ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X5 @ Y ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1955_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_int,Y: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X5: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1956_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [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 @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                   => ( ( 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_1957_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_1958_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [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 @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                   => ( ( 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_1959_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1960_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% count_le_length
thf(fact_1961_count__le__length,axiom,
    ! [Xs: list_int,X: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X ) @ ( size_size_list_int @ Xs ) ) ).

% count_le_length
thf(fact_1962_count__le__length,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_1963_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1964_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1965_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_1966_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_1967_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_1968_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_1969_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1970_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1971_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1972_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_1973_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_1974_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_1975_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_1976_power__mono__iff,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1977_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1978_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1979_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_1980_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_1981_power__eq__0__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1982_power__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( power_power_real @ A @ N2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1983_power__eq__0__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ( power_power_int @ A @ N2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1984_power__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( power_power_complex @ A @ N2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_1985_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_1986_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_1987_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X2: nat,N: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% high_def
thf(fact_1988__C9_C,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% "9"
thf(fact_1989_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1990_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1991_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1992_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1993_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1994_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1995_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1996_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1997_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1998_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1999_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2000_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2001_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2002_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2003_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_2004_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_2005_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( suc @ N2 ) )
      = zero_z5237406670263579293d_enat ) ).

% power_0_Suc
thf(fact_2006_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2007_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2008_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2009_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2010_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( numeral_numeral_nat @ K ) )
      = zero_z5237406670263579293d_enat ) ).

% power_zero_numeral
thf(fact_2011_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_2012_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_2013_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_2014_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_2015_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2016_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2017_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2018_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2019_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2020_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2021_nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_2022_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_2023_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_2024_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2025_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2026_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2027_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2028_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2029_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2030_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2031_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2032_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2033_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2034_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2035_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2036_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2037_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2038_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2039_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2040_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2041_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_2042_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_2043_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_real @ W2 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_2044_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2045_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2046_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2047_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2048_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2049_field__sum__of__halves,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_2050_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_2051_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2052_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2053_power__not__zero,axiom,
    ! [A: nat,N2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N2 )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_2054_power__not__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N2 )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_2055_power__not__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N2 )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_2056_power__not__zero,axiom,
    ! [A: complex,N2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N2 )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_2057_power__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2058_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2059_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2060_zero__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2061_zero__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2062_zero__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2063_zero__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2064_zero__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2065_zero__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2066_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_2067_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2068_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2069_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2070_power__inject__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N2 ) )
        = ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2071_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2072_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2073_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2074_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2075_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2076_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
        = zero_z5237406670263579293d_enat ) ) ).

% zero_power
thf(fact_2077_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_nat @ zero_zero_nat @ N2 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_2078_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ zero_zero_real @ N2 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_2079_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_int @ zero_zero_int @ N2 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_2080_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ zero_zero_complex @ N2 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_2081_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_2082_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_2083_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N2 )
              = ( power_power_real @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2084_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2085_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2086_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ N2 )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2087_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2088_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2089_zero__power2,axiom,
    ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z5237406670263579293d_enat ) ).

% zero_power2
thf(fact_2090_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2091_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2092_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2093_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2094_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_2095_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_2096_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_2097_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_2098_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2099_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2100_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2101_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2102_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2103_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2104_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2105_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2106_power__strict__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2107_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2108_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2109_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_2110_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_2111_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2112_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2113_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2114_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2115_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2116_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2117_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2118_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2119_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2120_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2121_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2122_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_2123_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_2124_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_2125_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_2126_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_2127_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_2128_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_2129_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_2130_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_2131_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2132_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2133_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2134_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2135_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2136_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2137_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_2138_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_2139_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_2140_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_2141_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_2142_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_2143_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2144_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2145_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_2146_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_2147_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_2148_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_2149_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B ) )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2150_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2151_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_2152_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_2153_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_2154_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_2155_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = one_one_complex )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2156_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2157_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2158_power__inject__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2159_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2160_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_2161_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_2162_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_2163_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_2164_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_2165_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_2166_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_2167_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_2168_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_2169_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(5)
thf(fact_2170_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_2171_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_2172_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_2173_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(6)
thf(fact_2174_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_2175_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_2176_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_2177_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_2178_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_2179_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_2180_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2181_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2182_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2183_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2184_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_2185_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2186_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2187_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2188_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2189_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2190_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2191_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2192_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2193_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2194_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2195_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2196_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2197_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2198_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2199_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2200_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2201_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2202_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2203_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_2204_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2205_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2206_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2207_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2208_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2209_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2210_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2211_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2212_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2213_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2214_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2215_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2216_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2217_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2218_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2219_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2220_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2221_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2222_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_2223_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_2224_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_2225_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_2226_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2227_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2228_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2229_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_2230_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_2231_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_2232_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_2233_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2234_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2235_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2236_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2237_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2238_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2239_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2240_zero__neq__one,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_neq_one
thf(fact_2241_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2242_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).

% less_numeral_extra(4)
thf(fact_2243_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_2244_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2245_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2246_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2247_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2248_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2249_not__one__le__zero,axiom,
    ~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).

% not_one_le_zero
thf(fact_2250_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2251_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2252_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2253_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2254_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_2255_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_2256_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_2257_zero__less__one__class_Ozero__le__one,axiom,
    ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% zero_less_one_class.zero_le_one
thf(fact_2258_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_2259_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_2260_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_2261_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2262_not__one__less__zero,axiom,
    ~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).

% not_one_less_zero
thf(fact_2263_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2264_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2265_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2266_zero__less__one,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% zero_less_one
thf(fact_2267_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2268_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2269_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2270_less__numeral__extra_I1_J,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% less_numeral_extra(1)
thf(fact_2271_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2272_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2273_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% one_le_numeral
thf(fact_2274_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_2275_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_2276_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_2277_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_2278_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_2279_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_2280_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_2281_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2282_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2283_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2284_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_2285_add__mono1,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ B @ one_on7984719198319812577d_enat ) ) ) ).

% add_mono1
thf(fact_2286_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_2287_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_2288_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = one_one_complex )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2289_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2290_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_2291_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_2292_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_2293_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_2294_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_2295_one__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2296_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2297_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2298_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N2 )
          = M )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2299_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_2300_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2301_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2302_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2303_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2304_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2305_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2306_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2307_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2308_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2309_zero__less__two,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).

% zero_less_two
thf(fact_2310_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2311_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2312_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2313_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2314_power__le__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2315_power__le__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2316_power__le__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2317_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_2318_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_2319_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = one_on7984719198319812577d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_8040749407984259932d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = zero_z5237406670263579293d_enat ) ) ) ).

% power_0_left
thf(fact_2320_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2321_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_2322_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2323_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_2324_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2325_power__gt1,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2326_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2327_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2328_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2329_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2330_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2331_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2332_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2333_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2334_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2335_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2336_nat__induct__non__zero,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P2 @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P2 @ N3 )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2337_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2338_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2339_power__Suc__le__self,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2340_power__Suc__le__self,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2341_power__Suc__le__self,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2342_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2343_power__Suc__less__one,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2344_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2345_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2346_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2347_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2348_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2349_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2350_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2351_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2352_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2353_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2354_self__le__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2355_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2356_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2357_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2358_one__less__power,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2359_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2360_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_2361_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_2362_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_2363_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_2364_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_2365_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2366_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_2367_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_2368_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_2369_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2370_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2371_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2372_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2373_nat__induct2,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ( P2 @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( P2 @ N3 )
             => ( P2 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_2374_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D4 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_2375_low__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = X ) ) ).

% low_inv
thf(fact_2376_high__inv,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
        = Y ) ) ).

% high_inv
thf(fact_2377_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_2378_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_2379_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2380_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2381_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2382_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2383_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_2384_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2385_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2386_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_2387_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_2388_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_2389_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_2390_mult__zero__left,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat @ A )
      = zero_z5237406670263579293d_enat ) ).

% mult_zero_left
thf(fact_2391_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_2392_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_2393_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_2394_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_2395_mult__zero__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% mult_zero_right
thf(fact_2396_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_2397_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_2398_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_2399_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_2400_mult__eq__0__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B )
        = zero_z5237406670263579293d_enat )
      = ( ( A = zero_z5237406670263579293d_enat )
        | ( B = zero_z5237406670263579293d_enat ) ) ) ).

% mult_eq_0_iff
thf(fact_2401_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2402_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2403_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2404_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( times_times_complex @ C @ A )
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2405_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2406_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2407_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2408_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( times_times_complex @ A @ C )
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2409_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_2410_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_2411_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_2412_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_2413_mult__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat @ A )
      = A ) ).

% mult_1
thf(fact_2414_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_2415_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_2416_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_2417_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_2418_mult_Oright__neutral,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ one_on7984719198319812577d_enat )
      = A ) ).

% mult.right_neutral
thf(fact_2419_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2420_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2421_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2422_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2423_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2424_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2425_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2426_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2427_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2428_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2429_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2430_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2431_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2432_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2433_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2434_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2435_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_2436_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_2437_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_2438_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_2439_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_2440_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2441_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_2442_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_2443_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_2444_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_2445_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2446_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2447_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2448_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2449_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2450_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2451_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2452_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2453_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2454_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2455_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2456_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2457_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2458_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2459_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2460_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2461_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2462_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2463_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2464_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2465_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2466_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2467_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2468_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2469_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2470_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2471_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2472_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2473_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2474_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2475_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2476_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2477_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2478_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2479_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2480_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2481_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2482_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2483_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_2484_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2485_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ M @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_2486_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2487_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2488_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2489_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2490_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2491_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2492_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2493_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2494_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2495_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2496_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2497_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2498_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2499_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2500_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2501_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2502_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2503_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2504_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2505_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2506_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_2507_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_2508_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_2509_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_2510_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2511_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2512_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2513_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2514_mult_Oleft__commute,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ B @ ( times_7803423173614009249d_enat @ A @ C ) )
      = ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2515_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2516_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2517_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2518_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2519_mult_Ocommute,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( times_7803423173614009249d_enat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2520_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2521_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2522_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2523_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2524_mult_Oassoc,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C )
      = ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2525_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2526_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2527_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2528_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2529_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C )
      = ( times_7803423173614009249d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2530_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_2531_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_2532_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2533_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_2534_mult__not__zero,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B )
       != zero_z5237406670263579293d_enat )
     => ( ( A != zero_z5237406670263579293d_enat )
        & ( B != zero_z5237406670263579293d_enat ) ) ) ).

% mult_not_zero
thf(fact_2535_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_2536_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_2537_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2538_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_2539_divisors__zero,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B )
        = zero_z5237406670263579293d_enat )
     => ( ( A = zero_z5237406670263579293d_enat )
        | ( B = zero_z5237406670263579293d_enat ) ) ) ).

% divisors_zero
thf(fact_2540_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_2541_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_2542_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2543_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( times_times_complex @ A @ B )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_2544_no__zero__divisors,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != zero_z5237406670263579293d_enat )
     => ( ( B != zero_z5237406670263579293d_enat )
       => ( ( times_7803423173614009249d_enat @ A @ B )
         != zero_z5237406670263579293d_enat ) ) ) ).

% no_zero_divisors
thf(fact_2545_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2546_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2547_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2548_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A )
          = ( times_times_complex @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2549_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2550_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2551_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2552_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = ( times_times_complex @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2553_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2554_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2555_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2556_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_2557_mult_Ocomm__neutral,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ one_on7984719198319812577d_enat )
      = A ) ).

% mult.comm_neutral
thf(fact_2558_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2559_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2560_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2561_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2562_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2563_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2564_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2565_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W2 @ Z ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2566_crossproduct__eq,axiom,
    ! [W2: complex,Y: complex,X: complex,Z: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y ) @ ( times_times_complex @ X @ Z ) )
        = ( plus_plus_complex @ ( times_times_complex @ W2 @ Z ) @ ( times_times_complex @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2567_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2568_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2569_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2570_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
       != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2571_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2572_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2573_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2574_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2575_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2576_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2577_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2578_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2579_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2580_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2581_comm__semiring__class_Odistrib,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2582_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2583_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_2584_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_2585_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% distrib_left
thf(fact_2586_distrib__left,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ ( times_7803423173614009249d_enat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2587_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2588_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_2589_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_2590_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% distrib_right
thf(fact_2591_distrib__right,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ C )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2592_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2593_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2594_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2595_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2596_combine__common__factor,axiom,
    ! [A: extended_enat,E2: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ E2 ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ B @ E2 ) @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2597_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2598_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2599_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2600_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2601_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W2: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2602_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W2: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2603_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_2604_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2605_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2606_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2607_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_2608_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_2609_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_2610_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_2611_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_2612_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_2613_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_2614_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_2615_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_2616_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2617_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2618_mult__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ C @ D )
       => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
           => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2619_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2620_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2621_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2622_mult__mono_H,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ C @ D )
       => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
           => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2623_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2624_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2625_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2626_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2627_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2628_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2629_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2630_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2631_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2632_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2633_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2634_mult__left__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ C @ A ) @ ( times_7803423173614009249d_enat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2635_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2636_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2637_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2638_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2639_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2640_mult__right__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2641_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2642_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2643_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2644_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2645_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2646_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2647_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2648_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2649_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2650_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2651_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2652_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2653_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2654_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2655_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2656_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2657_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2658_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2659_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2660_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2661_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2662_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2663_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ C @ A ) @ ( times_7803423173614009249d_enat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2664_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2665_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2666_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2667_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2668_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2669_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2670_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2671_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2672_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2673_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2674_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2675_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2676_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2677_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2678_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2679_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2680_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2681_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2682_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2683_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2684_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2685_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2686_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2687_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2688_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2689_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2690_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2691_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2692_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2693_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2694_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2695_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2696_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2697_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2698_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2699_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2700_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2701_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2702_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2703_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2704_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2705_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2706_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2707_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2708_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2709_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2710_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2711_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2712_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2713_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2714_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2715_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2716_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2717_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2718_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2719_less__1__mult,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N2 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2720_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2721_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2722_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2723_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B @ C )
          = A )
        = ( B
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2724_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2725_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B )
       => ( A
          = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2726_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2727_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2728_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2729_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_2730_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2731_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_2732_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2733_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W2 @ Z ) )
          = ( ( times_times_complex @ X @ Z )
            = ( times_times_complex @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2734_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W2 @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2735_power__Suc2,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2736_power__Suc2,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2737_power__Suc2,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2738_power__Suc2,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2739_power__Suc2,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_2740_power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2741_power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2742_power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2743_power__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2744_power__Suc,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ A @ ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_2745_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_2746_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2747_power__add,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_2748_power__add,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_add
thf(fact_2749_power__add,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_add
thf(fact_2750_power__add,axiom,
    ! [A: extended_enat,M: nat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A @ M ) @ ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2751_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_2752_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_2753_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_2754_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_2755_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2756_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_2757_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_2758_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_2759_power__odd__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2760_power__odd__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2761_power__odd__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2762_power__odd__eq,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2763_power__odd__eq,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ A @ ( power_8040749407984259932d_enat @ ( power_8040749407984259932d_enat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2764_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_2765_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_2766_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2767_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2768_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2769_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2770_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2771_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2772_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2773_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2774_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2775_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2776_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2777_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2778_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2779_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2780_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2781_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2782_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2783_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2784_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2785_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2786_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2787_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2788_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2789_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2790_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2791_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2792_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2793_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2794_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2795_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2796_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2797_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2798_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2799_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2800_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2801_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2802_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2803_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2804_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2805_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2806_mult__le__one,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ one_on7984719198319812577d_enat )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ( ord_le2932123472753598470d_enat @ B @ one_on7984719198319812577d_enat )
         => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ one_on7984719198319812577d_enat ) ) ) ) ).

% mult_le_one
thf(fact_2807_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2808_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_2809_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_2810_mult__left__le,axiom,
    ! [C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ one_on7984719198319812577d_enat )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
       => ( ord_le2932123472753598470d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2811_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2812_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2813_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2814_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2815_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2816_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2817_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2818_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2819_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2820_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2821_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2822_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2823_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2824_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2825_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2826_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2827_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2828_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2829_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2830_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2831_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2832_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W2 )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2833_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2834_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2835_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( 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_2836_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2837_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2838_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2839_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2840_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2841_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2842_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2843_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2844_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2845_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2846_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2847_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2848_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2849_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2850_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2851_power__less__power__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2852_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2853_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2854_power__gt1__lemma,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2855_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2856_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_2857_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2858_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2859_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2860_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_2861_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2862_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2863_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2864_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2865_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2866_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2867_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2868_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2869_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2870_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2871_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2872_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2873_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2874_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2875_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2876_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2877_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( ( ord_less_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2878_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2879_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2880_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2881_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2882_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2883_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2884_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2885_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2886_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2887_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2888_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2889_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2890_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( 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 ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2891_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ 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_2892_power__Suc__less,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2893_power__Suc__less,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2894_power__Suc__less,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2895_left__add__twice,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2896_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2897_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2898_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2899_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2900_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_2901_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_2902_mult__2__right,axiom,
    ! [Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_2903_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_2904_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_2905_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_2906_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_2907_mult__2,axiom,
    ! [Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
      = ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).

% mult_2
thf(fact_2908_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_2909_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_2910_div__nat__eqI,axiom,
    ! [N2: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_2911_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2912_split__div,axiom,
    ! [P2: nat > $o,M: nat,N2: nat] :
      ( ( P2 @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I5 ) @ J3 ) )
               => ( P2 @ I5 ) ) ) ) ) ) ).

% split_div
thf(fact_2913_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_2914_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2915_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2916_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2917_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( 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 ) @ B ) )
            & ( ~ ( 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_2918_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ 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_2919_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_2920_split__div_H,axiom,
    ! [P2: nat > $o,M: nat,N2: nat] :
      ( ( P2 @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P2 @ zero_zero_nat ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
            & ( P2 @ Q5 ) ) ) ) ).

% split_div'
thf(fact_2921_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2922_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2923_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2924_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2925_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_2926_zero__le__even__power_H,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2927_zero__le__even__power_H,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2928_nat__bit__induct,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P2 @ N3 )
           => ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
       => ( ! [N3: nat] :
              ( ( P2 @ N3 )
             => ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
         => ( P2 @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_2929_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2930_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_2931_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2932_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N2 )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N2 )
                    = A ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2933_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2934_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2935_odd__power__less__zero,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_2936_odd__power__less__zero,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_2937_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2938_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = ( divide_divide_nat @ M @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_2939_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2940_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_2941_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_2942_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_2943_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_2944_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2945_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2946_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2947_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2948_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2949_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2950_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2951_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_2952_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2953_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2954_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2955_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2956_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_2957_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_2958_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X2: nat,N: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% low_def
thf(fact_2959_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_2960_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_2961_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_2962_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_2963_set__decode__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( member_nat2 @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
      = ( member_nat2 @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_2964_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_2965_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2966_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2967_length__product,axiom,
    ! [Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_2968_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2969_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( size_s7647898544948552527nt_nat @ ( product_int_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2970_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_2971_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( size_s2970893825323803983at_int @ ( product_nat_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_2972_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_2973_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_2974_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_2975_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg1Leaf
thf(fact_2976_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A5: $o,B4: $o] :
          ( T
          = ( vEBT_Leaf @ A5 @ B4 ) ) ) ).

% deg_1_Leaf
thf(fact_2977_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A5: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).

% deg_1_Leafy
thf(fact_2978_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_2979_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_2980_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_2981_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_2982_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_2983_dvd__0__right,axiom,
    ! [A: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ zero_z5237406670263579293d_enat ) ).

% dvd_0_right
thf(fact_2984_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_2985_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_2986_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_2987_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_2988_dvd__0__left__iff,axiom,
    ! [A: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ zero_z5237406670263579293d_enat @ A )
      = ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left_iff
thf(fact_2989_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2990_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2991_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_2992_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2993_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2994_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_2995_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_2996_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_2997_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_2998_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_2999_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_3000_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_3001_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_3002_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_3003_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_3004_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_3005_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3006_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3007_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_3008_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_3009_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_3010_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_3011_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_3012_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_3013_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y2: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_3014_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y2: num] :
            ( X
            = ( some_num @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_3015_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y2: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y2 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_3016_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y2: num] :
            ( X
           != ( some_num @ Y2 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_3017_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3018_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3019_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3020_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3021_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3022_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3023_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3024_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3025_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3026_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3027_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_3028_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_3029_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3030_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3031_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3032_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3033_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3034_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3035_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3036_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3037_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_3038_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_3039_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_3040_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_3041_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_3042_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_3043_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_3044_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_3045_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3046_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3047_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3048_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3049_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3050_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3051_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_3052_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_3053_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_3054_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_3055_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3056_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3057_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_3058_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_3059_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_3060_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_3061_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3062_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3063_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3064_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3065_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3066_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3067_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3068_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3069_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_3070_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_3071_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_3072_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_3073_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3074_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3075_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3076_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3077_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_Suc_Suc_iff
thf(fact_3078_even__Suc,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% even_Suc
thf(fact_3079_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3080_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3081_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_3082_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_3083_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_3084_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_3085_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3086_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3087_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3088_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3089_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_3090_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_3091_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_3092_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_3093_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat2 @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_3094_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3095_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3096_power__less__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_3097_power__less__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_3098_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3099_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3100_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3101_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3102_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_3103_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_3104_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_3105_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_3106_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_3107_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_3108_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3109_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3110_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_3111_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_3112_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3113_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3114_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3115_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3116_even__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3117_even__power,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3118_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_3119_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3120_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3121_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3122_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3123_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3124_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3125_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3126_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3127_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( modulo_modulo_nat @ B3 @ A3 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3128_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A3: int,B3: int] :
          ( ( modulo_modulo_int @ B3 @ A3 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3129_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3130_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3131_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3132_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3133_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3134_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3135_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3136_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3137_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_3138_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_3139_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_3140_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_3141_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_3142_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_3143_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_3144_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_3145_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3146_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3147_mod__add__cong,axiom,
    ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A6 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B6 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3148_mod__add__cong,axiom,
    ! [A: int,C: int,A6: int,B: int,B6: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A6 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B6 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3149_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3150_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3151_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3152_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3153_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B3: real] :
          ( ( A3 = zero_zero_real )
         => ( B3 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_3154_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B3: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B3 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_3155_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_3156_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_3157_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_3158_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_3159_dvd__0__left,axiom,
    ! [A: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( A = zero_z5237406670263579293d_enat ) ) ).

% dvd_0_left
thf(fact_3160_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_3161_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_3162_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_3163_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_3164_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_3165_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3166_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3167_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3168_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3169_dvd__triv__right,axiom,
    ! [A: extended_enat,B: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3170_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3171_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3172_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3173_dvd__mult__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
     => ( dvd_dvd_complex @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3174_dvd__mult__right,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C )
     => ( dvd_dv3785147216227455552d_enat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3175_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3176_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3177_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3178_mult__dvd__mono,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( ( dvd_dvd_complex @ C @ D )
       => ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3179_mult__dvd__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ B )
     => ( ( dvd_dv3785147216227455552d_enat @ C @ D )
       => ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A @ C ) @ ( times_7803423173614009249d_enat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3180_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3181_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3182_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3183_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3184_dvd__triv__left,axiom,
    ! [A: extended_enat,B: extended_enat] : ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3185_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3186_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3187_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3188_dvd__mult__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
     => ( dvd_dvd_complex @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3189_dvd__mult__left,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ ( times_7803423173614009249d_enat @ A @ B ) @ C )
     => ( dvd_dv3785147216227455552d_enat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3190_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3191_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3192_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3193_dvd__mult2,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3194_dvd__mult2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ B )
     => ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3195_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3196_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3197_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3198_dvd__mult,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ C )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3199_dvd__mult,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ C )
     => ( dvd_dv3785147216227455552d_enat @ A @ ( times_7803423173614009249d_enat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3200_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A3: nat] :
        ? [K2: nat] :
          ( A3
          = ( times_times_nat @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3201_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B3: int,A3: int] :
        ? [K2: int] :
          ( A3
          = ( times_times_int @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3202_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B3: real,A3: real] :
        ? [K2: real] :
          ( A3
          = ( times_times_real @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3203_dvd__def,axiom,
    ( dvd_dvd_complex
    = ( ^ [B3: complex,A3: complex] :
        ? [K2: complex] :
          ( A3
          = ( times_times_complex @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3204_dvd__def,axiom,
    ( dvd_dv3785147216227455552d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
        ? [K2: extended_enat] :
          ( A3
          = ( times_7803423173614009249d_enat @ B3 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_3205_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_3206_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_3207_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_3208_dvdI,axiom,
    ! [A: complex,B: complex,K: complex] :
      ( ( A
        = ( times_times_complex @ B @ K ) )
     => ( dvd_dvd_complex @ B @ A ) ) ).

% dvdI
thf(fact_3209_dvdI,axiom,
    ! [A: extended_enat,B: extended_enat,K: extended_enat] :
      ( ( A
        = ( times_7803423173614009249d_enat @ B @ K ) )
     => ( dvd_dv3785147216227455552d_enat @ B @ A ) ) ).

% dvdI
thf(fact_3210_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_3211_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B @ K3 ) ) ) ).

% dvdE
thf(fact_3212_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B @ K3 ) ) ) ).

% dvdE
thf(fact_3213_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ~ ! [K3: complex] :
            ( A
           != ( times_times_complex @ B @ K3 ) ) ) ).

% dvdE
thf(fact_3214_dvdE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ B @ A )
     => ~ ! [K3: extended_enat] :
            ( A
           != ( times_7803423173614009249d_enat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_3215_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3216_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3217_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3218_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3219_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_3220_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_3221_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_3222_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_3223_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3224_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3225_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3226_dvd__add,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( dvd_dv3785147216227455552d_enat @ A @ B )
     => ( ( dvd_dv3785147216227455552d_enat @ A @ C )
       => ( dvd_dv3785147216227455552d_enat @ A @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3227_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3228_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3229_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3230_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3231_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3232_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3233_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_3234_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_3235_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3236_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3237_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3238_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3239_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3240_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3241_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_3242_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_3243_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_3244_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_3245_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_3246_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_3247_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_3248_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_3249_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_3250_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_3251_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_3252_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_3253_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P: option4927543243414619207at_nat > $o] :
          ( ( P @ none_P5556105721700978146at_nat )
          | ? [X2: product_prod_nat_nat] : ( P @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_3254_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P: option_num > $o] :
          ( ( P @ none_num )
          | ? [X2: num] : ( P @ ( some_num @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_3255_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P: option4927543243414619207at_nat > $o] :
          ( ( P @ none_P5556105721700978146at_nat )
          & ! [X2: product_prod_nat_nat] : ( P @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_3256_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P: option_num > $o] :
          ( ( P @ none_num )
          & ! [X2: num] : ( P @ ( some_num @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_3257_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B4 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3258_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B4: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B4 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3259_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A5: num,B4: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B4 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3260_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A5: num,B4: num] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_num @ B4 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3261_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_3262_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_3263_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_3264_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3265_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_3266_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_3267_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_3268_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3269_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3270_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3271_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3272_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3273_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3274_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D5: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_3275_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3276_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3277_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3278_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3279_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3280_pinf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ( ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) )
        = ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3281_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3282_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3283_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3284_pinf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ X3 )
     => ( ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) )
        = ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3285_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3286_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z2 @ X3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3287_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3288_minf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ( ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) )
        = ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3289_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3290_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3291_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3292_minf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z2: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z2 )
     => ( ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) )
        = ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3293_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3294_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3295_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3296_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3297_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3298_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( ( divide_divide_real @ A @ B )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3299_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_3300_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_3301_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3302_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3303_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3304_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3305_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3306_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3307_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3308_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3309_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3310_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3311_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3312_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3313_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
          = N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
         != N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_3314_mod__induct,axiom,
    ! [P2: nat > $o,N2: nat,P4: nat,M: nat] :
      ( ( P2 @ N2 )
     => ( ( ord_less_nat @ N2 @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P4 )
               => ( ( P2 @ N3 )
                 => ( P2 @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
           => ( P2 @ M ) ) ) ) ) ).

% mod_induct
thf(fact_3315_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3316_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3317_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3318_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3319_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3320_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3321_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3322_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3323_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3324_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3325_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3326_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3327_gcd__nat__induct,axiom,
    ! [P2: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M3: nat] : ( P2 @ M3 @ zero_zero_nat )
     => ( ! [M3: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P2 @ N3 @ ( modulo_modulo_nat @ M3 @ N3 ) )
             => ( P2 @ M3 @ N3 ) ) )
       => ( P2 @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_3328_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_3329_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3330_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3331_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3332_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3333_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3334_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3335_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3336_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3337_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3338_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3339_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_3340_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_3341_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3342_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3343_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3344_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3345_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3346_power__le__dvd,axiom,
    ! [A: real,N2: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3347_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3348_power__le__dvd,axiom,
    ! [A: complex,N2: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3349_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3350_dvd__power__le,axiom,
    ! [X: real,Y: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3351_dvd__power__le,axiom,
    ! [X: int,Y: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3352_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3353_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3354_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_3355_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_3356_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X5: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3357_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X5: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) )
        | ( ( times_times_nat @ B @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_nat
thf(fact_3358_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3359_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_3360_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_3361_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_3362_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_3363_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_3364_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).

% vebt_member.simps(2)
thf(fact_3365_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B4: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X5 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_3366_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B4: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
            ( X
           != ( 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] :
              ( X
             != ( 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] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_3367_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( 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] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X5 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3368_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_3369_find__None__iff,axiom,
    ! [P2: real > $o,Xs: list_real] :
      ( ( ( find_real @ P2 @ Xs )
        = none_real )
      = ( ~ ? [X2: real] :
              ( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3370_find__None__iff,axiom,
    ! [P2: $o > $o,Xs: list_o] :
      ( ( ( find_o @ P2 @ Xs )
        = none_o )
      = ( ~ ? [X2: $o] :
              ( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3371_find__None__iff,axiom,
    ! [P2: set_nat > $o,Xs: list_set_nat] :
      ( ( ( find_set_nat @ P2 @ Xs )
        = none_set_nat )
      = ( ~ ? [X2: set_nat] :
              ( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3372_find__None__iff,axiom,
    ! [P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P2 @ Xs )
        = none_VEBT_VEBT )
      = ( ~ ? [X2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3373_find__None__iff,axiom,
    ! [P2: int > $o,Xs: list_int] :
      ( ( ( find_int @ P2 @ Xs )
        = none_int )
      = ( ~ ? [X2: int] :
              ( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3374_find__None__iff,axiom,
    ! [P2: nat > $o,Xs: list_nat] :
      ( ( ( find_nat @ P2 @ Xs )
        = none_nat )
      = ( ~ ? [X2: nat] :
              ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3375_find__None__iff,axiom,
    ! [P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P2 @ Xs )
        = none_P5556105721700978146at_nat )
      = ( ~ ? [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3376_find__None__iff,axiom,
    ! [P2: num > $o,Xs: list_num] :
      ( ( ( find_num @ P2 @ Xs )
        = none_num )
      = ( ~ ? [X2: num] :
              ( ( member_num @ X2 @ ( set_num2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff
thf(fact_3377_find__None__iff2,axiom,
    ! [P2: real > $o,Xs: list_real] :
      ( ( none_real
        = ( find_real @ P2 @ Xs ) )
      = ( ~ ? [X2: real] :
              ( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3378_find__None__iff2,axiom,
    ! [P2: $o > $o,Xs: list_o] :
      ( ( none_o
        = ( find_o @ P2 @ Xs ) )
      = ( ~ ? [X2: $o] :
              ( ( member_o2 @ X2 @ ( set_o2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3379_find__None__iff2,axiom,
    ! [P2: set_nat > $o,Xs: list_set_nat] :
      ( ( none_set_nat
        = ( find_set_nat @ P2 @ Xs ) )
      = ( ~ ? [X2: set_nat] :
              ( ( member_set_nat2 @ X2 @ ( set_set_nat2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3380_find__None__iff2,axiom,
    ! [P2: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT
        = ( find_VEBT_VEBT @ P2 @ Xs ) )
      = ( ~ ? [X2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3381_find__None__iff2,axiom,
    ! [P2: int > $o,Xs: list_int] :
      ( ( none_int
        = ( find_int @ P2 @ Xs ) )
      = ( ~ ? [X2: int] :
              ( ( member_int2 @ X2 @ ( set_int2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3382_find__None__iff2,axiom,
    ! [P2: nat > $o,Xs: list_nat] :
      ( ( none_nat
        = ( find_nat @ P2 @ Xs ) )
      = ( ~ ? [X2: nat] :
              ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3383_find__None__iff2,axiom,
    ! [P2: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat
        = ( find_P8199882355184865565at_nat @ P2 @ Xs ) )
      = ( ~ ? [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3384_find__None__iff2,axiom,
    ! [P2: num > $o,Xs: list_num] :
      ( ( none_num
        = ( find_num @ P2 @ Xs ) )
      = ( ~ ? [X2: num] :
              ( ( member_num @ X2 @ ( set_num2 @ Xs ) )
              & ( P2 @ X2 ) ) ) ) ).

% find_None_iff2
thf(fact_3385_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3386_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3387_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3388_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3389_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3390_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3391_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_3392_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_3393_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B4: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ X5 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3394_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3395_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3396_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3397_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3398_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3399_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3400_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3401_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3402_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3403_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3404_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3405_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3406_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3407_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3408_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3409_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3410_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_3411_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_3412_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_3413_unity__coeff__ex,axiom,
    ! [P2: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P2 @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3414_unity__coeff__ex,axiom,
    ! [P2: int > $o,L: int] :
      ( ( ? [X2: int] : ( P2 @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3415_unity__coeff__ex,axiom,
    ! [P2: real > $o,L: real] :
      ( ( ? [X2: real] : ( P2 @ ( times_times_real @ L @ X2 ) ) )
      = ( ? [X2: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X2 @ zero_zero_real ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3416_unity__coeff__ex,axiom,
    ! [P2: complex > $o,L: complex] :
      ( ( ? [X2: complex] : ( P2 @ ( times_times_complex @ L @ X2 ) ) )
      = ( ? [X2: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3417_unity__coeff__ex,axiom,
    ! [P2: extended_enat > $o,L: extended_enat] :
      ( ( ? [X2: extended_enat] : ( P2 @ ( times_7803423173614009249d_enat @ L @ X2 ) ) )
      = ( ? [X2: extended_enat] :
            ( ( dvd_dv3785147216227455552d_enat @ L @ ( plus_p3455044024723400733d_enat @ X2 @ zero_z5237406670263579293d_enat ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3418_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3419_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3420_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3421_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3422_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3423_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3424_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3425_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3426_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3427_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3428_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3429_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3430_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3431_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3432_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3433_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3434_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3435_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3436_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3437_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3438_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3439_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3440_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_3441_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3442_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3443_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_3444_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B2 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_3445_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q3: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3446_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S2: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3447_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3448_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3449_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_3450_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_3451_div__mod__decomp,axiom,
    ! [A2: nat,N2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_3452_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_3453_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_3454_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_3455_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_3456_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X5: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D5 @ A )
          & ( dvd_dvd_nat @ D5 @ B )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3457_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3458_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_3459_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_3460_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_3461_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_3462_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_3463_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3464_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3465_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3466_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3467_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B4: nat] :
              ( ( B4 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B4 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B4 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3468_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B4: int] :
              ( ( B4 != zero_zero_int )
             => ( ( dvd_dvd_int @ B4 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B4 )
                 => ( ( ( divide_divide_int @ one_one_int @ B4 )
                      = A )
                   => ( ( ( times_times_int @ A @ B4 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3469_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_3470_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_3471_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3472_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3473_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3474_dvd__power__iff,axiom,
    ! [X: int,M: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3475_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3476_dvd__power,axiom,
    ! [N2: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3477_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3478_dvd__power,axiom,
    ! [N2: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3479_split__mod,axiom,
    ! [P2: nat > $o,M: nat,N2: nat] :
      ( ( P2 @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P2 @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I5 ) @ J3 ) )
               => ( P2 @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_3480_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_3481_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_3482_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_3483_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3484_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3485_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_Extended_enat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s3941691890525107288d_enat @ Ys ) ) )
     => ( ( nth_Pr7509392720524132704d_enat @ ( produc4894985897562121079d_enat @ Xs @ Ys ) @ N2 )
        = ( produc581526299967858633d_enat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s3941691890525107288d_enat @ Ys ) ) ) @ ( nth_Extended_enat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s3941691890525107288d_enat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3486_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3487_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3488_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N2 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3489_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3490_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys ) @ N2 )
        = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3491_product__nth,axiom,
    ! [N2: nat,Xs: list_int,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys ) @ N2 )
        = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3492_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3493_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys ) @ N2 )
        = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3494_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N2 )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3495_power__mono__odd,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_3496_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_3497_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_3498_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% odd_pos
thf(fact_3499_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_3500_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_3501_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_3502_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_3503_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_3504_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_3505_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_3506_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3507_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_3508_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_3509_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_3510_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3511_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3512_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_3513_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_3514_zero__le__even__power,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_3515_zero__le__even__power,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_3516_zero__le__odd__power,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_3517_zero__le__odd__power,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_3518_zero__le__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_3519_zero__le__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_3520_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B2 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_3521_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3522_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_3523_zero__less__power__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3524_zero__less__power__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3525_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3526_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3527_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3528_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3529_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3530_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3531_power__le__zero__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3532_power__le__zero__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3533_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3534_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3535_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A3: $o,B3: $o] :
                ( A1
                = ( vEBT_Leaf @ A3 @ B3 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X2 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X2 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X2 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ! [I5: nat] :
                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I5 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I5 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N )
                              = I5 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X2 @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I5: nat] :
                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I5 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I5 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N )
                              = I5 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3536_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A5: $o,B4: $o] :
              ( A12
              = ( vEBT_Leaf @ A5 @ B4 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X3 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                     => ( ( M3 = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M3 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X3 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                       => ( ( M3
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M3 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X3 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                         => ( ( M3 = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M3 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X3: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X3 )
                                                      & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X3 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                           => ( ( M3
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M3 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X3: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X3 )
                                                        & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3537_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3538_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3539_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_3540_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B4 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_3541_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3542_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3543_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_3544_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_3545_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3546_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3547_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3548_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3549_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3550_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3551_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_3552_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_3553_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3554_Diff__cancel,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ A2 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_3555_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3556_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3557_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3558_empty__Diff,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_3559_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3560_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3561_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_3562_Diff__empty,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% Diff_empty
thf(fact_3563_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_3564_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_3565_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3566_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3567_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3568_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3569_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_3570_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3571_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3572_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_3573_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3574_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_3575_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3576_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_3577_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_3578_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3579_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_3580_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3581_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3582_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3583_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3584_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3585_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3586_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3587_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3588_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3589_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3590_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3591_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3592_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3593_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3594_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3595_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3596_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3597_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( minus_minus_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3598_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3599_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3600_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_3601_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_3602_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3603_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3604_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_3605_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_3606_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_3607_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_3608_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_3609_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P2 )
        = zero_zero_real )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_3610_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P2 )
        = zero_zero_complex )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_3611_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n1046097342994218471d_enat @ P2 )
        = zero_z5237406670263579293d_enat )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_3612_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = zero_zero_nat )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_3613_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = zero_zero_int )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_3614_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_3615_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_3616_of__bool__eq_I1_J,axiom,
    ( ( zero_n1046097342994218471d_enat @ $false )
    = zero_z5237406670263579293d_enat ) ).

% of_bool_eq(1)
thf(fact_3617_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_3618_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_3619_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_le72135733267957522d_enat @ ( zero_n1046097342994218471d_enat @ P2 ) @ ( zero_n1046097342994218471d_enat @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3620_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3621_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3622_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_3623_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_3624_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_3625_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_3626_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_3627_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P2 )
        = one_one_real )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_3628_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P2 )
        = one_one_complex )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_3629_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = one_one_nat )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_3630_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = one_one_int )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_3631_signed__take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_3632_of__bool__or__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P2
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_3633_of__bool__or__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P2
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_3634_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3635_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3636_finite__Collect__subsets,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite5468666774076196335d_enat
        @ ( collec2260605976452661553d_enat
          @ ^ [B5: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3637_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3638_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_3639_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_3640_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3641_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3642_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3643_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3644_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3645_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3646_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_3647_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3648_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3649_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_3650_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_3651_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3652_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_3653_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_3654_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_3655_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_3656_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_3657_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_3658_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_3659_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_3660_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_3661_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_3662_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_3663_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_3664_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_3665_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P2 )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_3666_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P2 )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_3667_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P2 )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_3668_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_3669_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_3670_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_3671_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_3672_Suc__0__mod__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N2
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_3673_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_3674_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_3675_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_3676_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_3677_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_3678_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_3679_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_3680_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_3681_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_3682_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_3683_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% odd_Suc_minus_one
thf(fact_3684_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_3685_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3686_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3687_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3688_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3689_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3690_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3691_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3692_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3693_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_3694_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_3695_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X2: real] : ( member_real2 @ X2 @ R )
        @ ^ [X2: real] : ( member_real2 @ X2 @ S3 ) )
      = ( ord_less_eq_set_real @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_3696_pred__subset__eq,axiom,
    ! [R: set_o,S3: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X2: $o] : ( member_o2 @ X2 @ R )
        @ ^ [X2: $o] : ( member_o2 @ X2 @ S3 ) )
      = ( ord_less_eq_set_o @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_3697_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ R )
        @ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ S3 ) )
      = ( ord_le6893508408891458716et_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_3698_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X2: int] : ( member_int2 @ X2 @ R )
        @ ^ [X2: int] : ( member_int2 @ X2 @ S3 ) )
      = ( ord_less_eq_set_int @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_3699_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X2: nat] : ( member_nat2 @ X2 @ R )
        @ ^ [X2: nat] : ( member_nat2 @ X2 @ S3 ) )
      = ( ord_less_eq_set_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_3700_prop__restrict,axiom,
    ! [X: $o,Z5: set_o,X7: set_o,P2: $o > $o] :
      ( ( member_o2 @ X @ Z5 )
     => ( ( ord_less_eq_set_o @ Z5
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ X7 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_3701_prop__restrict,axiom,
    ! [X: real,Z5: set_real,X7: set_real,P2: real > $o] :
      ( ( member_real2 @ X @ Z5 )
     => ( ( ord_less_eq_set_real @ Z5
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ X7 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_3702_prop__restrict,axiom,
    ! [X: list_nat,Z5: set_list_nat,X7: set_list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ X @ Z5 )
     => ( ( ord_le6045566169113846134st_nat @ Z5
          @ ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( member_list_nat @ X2 @ X7 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_3703_prop__restrict,axiom,
    ! [X: set_nat,Z5: set_set_nat,X7: set_set_nat,P2: set_nat > $o] :
      ( ( member_set_nat2 @ X @ Z5 )
     => ( ( ord_le6893508408891458716et_nat @ Z5
          @ ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( member_set_nat2 @ X2 @ X7 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_3704_prop__restrict,axiom,
    ! [X: int,Z5: set_int,X7: set_int,P2: int > $o] :
      ( ( member_int2 @ X @ Z5 )
     => ( ( ord_less_eq_set_int @ Z5
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ X7 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_3705_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X7: set_nat,P2: nat > $o] :
      ( ( member_nat2 @ X @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat2 @ X2 @ X7 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_3706_Collect__restrict,axiom,
    ! [X7: set_o,P2: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X2: $o] :
            ( ( member_o2 @ X2 @ X7 )
            & ( P2 @ X2 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_3707_Collect__restrict,axiom,
    ! [X7: set_real,P2: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X2: real] :
            ( ( member_real2 @ X2 @ X7 )
            & ( P2 @ X2 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_3708_Collect__restrict,axiom,
    ! [X7: set_list_nat,P2: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X2: list_nat] :
            ( ( member_list_nat @ X2 @ X7 )
            & ( P2 @ X2 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_3709_Collect__restrict,axiom,
    ! [X7: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X2: set_nat] :
            ( ( member_set_nat2 @ X2 @ X7 )
            & ( P2 @ X2 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_3710_Collect__restrict,axiom,
    ! [X7: set_int,P2: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X2: int] :
            ( ( member_int2 @ X2 @ X7 )
            & ( P2 @ X2 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_3711_Collect__restrict,axiom,
    ! [X7: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat2 @ X2 @ X7 )
            & ( P2 @ X2 ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_3712_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A4: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X2: real] : ( member_real2 @ X2 @ A4 )
          @ ^ [X2: real] : ( member_real2 @ X2 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3713_less__eq__set__def,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A4: set_o,B5: set_o] :
          ( ord_less_eq_o_o
          @ ^ [X2: $o] : ( member_o2 @ X2 @ A4 )
          @ ^ [X2: $o] : ( member_o2 @ X2 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3714_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B5: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ A4 )
          @ ^ [X2: set_nat] : ( member_set_nat2 @ X2 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3715_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X2: int] : ( member_int2 @ X2 @ A4 )
          @ ^ [X2: int] : ( member_int2 @ X2 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3716_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X2: nat] : ( member_nat2 @ X2 @ A4 )
          @ ^ [X2: nat] : ( member_nat2 @ X2 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_3717_Collect__subset,axiom,
    ! [A2: set_o,P2: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X2: $o] :
            ( ( member_o2 @ X2 @ A2 )
            & ( P2 @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3718_Collect__subset,axiom,
    ! [A2: set_real,P2: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X2: real] :
            ( ( member_real2 @ X2 @ A2 )
            & ( P2 @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3719_Collect__subset,axiom,
    ! [A2: set_list_nat,P2: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X2: list_nat] :
            ( ( member_list_nat @ X2 @ A2 )
            & ( P2 @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3720_Collect__subset,axiom,
    ! [A2: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X2: set_nat] :
            ( ( member_set_nat2 @ X2 @ A2 )
            & ( P2 @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3721_Collect__subset,axiom,
    ! [A2: set_int,P2: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X2: int] :
            ( ( member_int2 @ X2 @ A2 )
            & ( P2 @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3722_Collect__subset,axiom,
    ! [A2: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat2 @ X2 @ A2 )
            & ( P2 @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3723_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P4 )
        = ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_3724_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P4 )
        = ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_3725_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_3726_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3727_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_3728_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_3729_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_3730_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_3731_empty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X2: list_nat] : $false ) ) ).

% empty_def
thf(fact_3732_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X2: set_nat] : $false ) ) ).

% empty_def
thf(fact_3733_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X2: real] : $false ) ) ).

% empty_def
thf(fact_3734_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X2: $o] : $false ) ) ).

% empty_def
thf(fact_3735_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $false ) ) ).

% empty_def
thf(fact_3736_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X2: int] : $false ) ) ).

% empty_def
thf(fact_3737_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_3738_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_3739_pred__equals__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ( ( ^ [X2: vEBT_VEBT,Y2: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: vEBT_VEBT,Y2: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X2 @ Y2 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_3740_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_3741_pred__equals__eq2,axiom,
    ! [R: set_Pr2457182780427864761d_enat,S3: set_Pr2457182780427864761d_enat] :
      ( ( ( ^ [X2: vEBT_VEBT,Y2: extended_enat] : ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X2 @ Y2 ) @ R ) )
        = ( ^ [X2: vEBT_VEBT,Y2: extended_enat] : ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X2 @ Y2 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_3742_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_3743_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_3744_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_3745_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_3746_lambda__zero,axiom,
    ( ( ^ [H: extended_enat] : zero_z5237406670263579293d_enat )
    = ( times_7803423173614009249d_enat @ zero_z5237406670263579293d_enat ) ) ).

% lambda_zero
thf(fact_3747_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_3748_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_3749_lambda__one,axiom,
    ( ( ^ [X2: real] : X2 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_3750_lambda__one,axiom,
    ( ( ^ [X2: complex] : X2 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_3751_lambda__one,axiom,
    ( ( ^ [X2: extended_enat] : X2 )
    = ( times_7803423173614009249d_enat @ one_on7984719198319812577d_enat ) ) ).

% lambda_one
thf(fact_3752_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3753_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3754_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3755_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3756_max__def__raw,axiom,
    ( ord_max_filter_nat
    = ( ^ [A3: filter_nat,B3: filter_nat] : ( if_filter_nat @ ( ord_le2510731241096832064er_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3757_max__def__raw,axiom,
    ( ord_max_real
    = ( ^ [A3: real,B3: real] : ( if_real @ ( ord_less_eq_real @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3758_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3759_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3760_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_3761_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_3762_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ S3 ) )
      = ( ord_le3000389064537975527at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_3763_pred__subset__eq2,axiom,
    ! [R: set_Pr7556676689462069481BT_nat,S3: set_Pr7556676689462069481BT_nat] :
      ( ( ord_le1182472622972956176_nat_o
        @ ^ [X2: vEBT_VEBT,Y2: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X2 @ Y2 ) @ R )
        @ ^ [X2: vEBT_VEBT,Y2: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X2 @ Y2 ) @ S3 ) )
      = ( ord_le3442269383143156041BT_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_3764_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R )
        @ ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ S3 ) )
      = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_3765_pred__subset__eq2,axiom,
    ! [R: set_Pr2457182780427864761d_enat,S3: set_Pr2457182780427864761d_enat] :
      ( ( ord_le2691948842708570076enat_o
        @ ^ [X2: vEBT_VEBT,Y2: extended_enat] : ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X2 @ Y2 ) @ R )
        @ ^ [X2: vEBT_VEBT,Y2: extended_enat] : ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X2 @ Y2 ) @ S3 ) )
      = ( ord_le8566326065971749465d_enat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_3766_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( ( dvd_dvd_real @ A @ B )
        & ~ ( dvd_dvd_real @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3767_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3768_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3769_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X2: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_3770_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_3771_bot__empty__eq2,axiom,
    ( bot_bo1565574316222977092_nat_o
    = ( ^ [X2: vEBT_VEBT,Y2: nat] : ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X2 @ Y2 ) @ bot_bo1642239108664514429BT_nat ) ) ) ).

% bot_empty_eq2
thf(fact_3772_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_3773_bot__empty__eq2,axiom,
    ( bot_bo2578006069712851624enat_o
    = ( ^ [X2: vEBT_VEBT,Y2: extended_enat] : ( member38198578724832770d_enat @ ( produc581526299967858633d_enat @ X2 @ Y2 ) @ bot_bo4330027929424010533d_enat ) ) ) ).

% bot_empty_eq2
thf(fact_3774_finite__M__bounded__by__nat,axiom,
    ! [P2: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P2 @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3775_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3776_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P2
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_3777_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n1201886186963655149omplex
        @ ( P2
          & Q ) )
      = ( times_times_complex @ ( zero_n1201886186963655149omplex @ P2 ) @ ( zero_n1201886186963655149omplex @ Q ) ) ) ).

% of_bool_conj
thf(fact_3778_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n1046097342994218471d_enat
        @ ( P2
          & Q ) )
      = ( times_7803423173614009249d_enat @ ( zero_n1046097342994218471d_enat @ P2 ) @ ( zero_n1046097342994218471d_enat @ Q ) ) ) ).

% of_bool_conj
thf(fact_3779_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P2
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_3780_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P2
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_3781_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3782_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3783_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_3784_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_3785_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_3786_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_3787_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_3788_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_3789_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: complex,Z3: complex] : Y5 = Z3 )
    = ( ^ [A3: complex,B3: complex] :
          ( ( minus_minus_complex @ A3 @ B3 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3790_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3791_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z3: real] : Y5 = Z3 )
    = ( ^ [A3: real,B3: real] :
          ( ( minus_minus_real @ A3 @ B3 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3792_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3793_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3794_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_3795_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_3796_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3797_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_3798_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3799_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_3800_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3801_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3802_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3803_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3804_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3805_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3806_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3807_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3808_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3809_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3810_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3811_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3812_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3813_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3814_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3815_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3816_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3817_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3818_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3819_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3820_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3821_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3822_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3823_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_3824_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3825_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3826_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3827_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_3828_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3829_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_3830_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3831_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_3832_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3833_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3834_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_3835_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_3836_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_3837_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_3838_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_3839_zero__induct__lemma,axiom,
    ! [P2: nat > $o,K: nat,I: nat] :
      ( ( P2 @ K )
     => ( ! [N3: nat] :
            ( ( P2 @ ( suc @ N3 ) )
           => ( P2 @ N3 ) )
       => ( P2 @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_3840_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3841_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_3842_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D6: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D6 @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D6 ) ) ) ) ).

% Diff_mono
thf(fact_3843_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_3844_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_3845_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_3846_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_3847_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_3848_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_3849_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_3850_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_3851_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_3852_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_3853_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_3854_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_3855_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_3856_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_3857_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_3858_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_3859_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_3860_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_3861_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_3862_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_3863_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_3864_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3865_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3866_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3867_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3868_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3869_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_3870_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_3871_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_3872_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_3873_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_3874_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_3875_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_3876_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_3877_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_3878_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_3879_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_3880_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_3881_of__bool__def,axiom,
    ( zero_n1046097342994218471d_enat
    = ( ^ [P6: $o] : ( if_Extended_enat @ P6 @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ) ) ).

% of_bool_def
thf(fact_3882_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_3883_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_3884_split__of__bool,axiom,
    ! [P2: real > $o,P4: $o] :
      ( ( P2 @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ( P4
         => ( P2 @ one_one_real ) )
        & ( ~ P4
         => ( P2 @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_3885_split__of__bool,axiom,
    ! [P2: complex > $o,P4: $o] :
      ( ( P2 @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ( P4
         => ( P2 @ one_one_complex ) )
        & ( ~ P4
         => ( P2 @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_3886_split__of__bool,axiom,
    ! [P2: extended_enat > $o,P4: $o] :
      ( ( P2 @ ( zero_n1046097342994218471d_enat @ P4 ) )
      = ( ( P4
         => ( P2 @ one_on7984719198319812577d_enat ) )
        & ( ~ P4
         => ( P2 @ zero_z5237406670263579293d_enat ) ) ) ) ).

% split_of_bool
thf(fact_3887_split__of__bool,axiom,
    ! [P2: nat > $o,P4: $o] :
      ( ( P2 @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ( P4
         => ( P2 @ one_one_nat ) )
        & ( ~ P4
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_3888_split__of__bool,axiom,
    ! [P2: int > $o,P4: $o] :
      ( ( P2 @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ( P4
         => ( P2 @ one_one_int ) )
        & ( ~ P4
         => ( P2 @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_3889_split__of__bool__asm,axiom,
    ! [P2: real > $o,P4: $o] :
      ( ( P2 @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P2 @ one_one_real ) )
            | ( ~ P4
              & ~ ( P2 @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3890_split__of__bool__asm,axiom,
    ! [P2: complex > $o,P4: $o] :
      ( ( P2 @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P2 @ one_one_complex ) )
            | ( ~ P4
              & ~ ( P2 @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3891_split__of__bool__asm,axiom,
    ! [P2: extended_enat > $o,P4: $o] :
      ( ( P2 @ ( zero_n1046097342994218471d_enat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P2 @ one_on7984719198319812577d_enat ) )
            | ( ~ P4
              & ~ ( P2 @ zero_z5237406670263579293d_enat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3892_split__of__bool__asm,axiom,
    ! [P2: nat > $o,P4: $o] :
      ( ( P2 @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P2 @ one_one_nat ) )
            | ( ~ P4
              & ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3893_split__of__bool__asm,axiom,
    ! [P2: int > $o,P4: $o] :
      ( ( P2 @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P2 @ one_one_int ) )
            | ( ~ P4
              & ~ ( P2 @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3894_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3895_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3896_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3897_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3898_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3899_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_3900_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3901_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_3902_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_3903_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_3904_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3905_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3906_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3907_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3908_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3909_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3910_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3911_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3912_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3913_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3914_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3915_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N2: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
         => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3916_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3917_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3918_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3919_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3920_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3921_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3922_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3923_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3924_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3925_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3926_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3927_square__diff__square__factored,axiom,
    ! [X: complex,Y: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ ( times_times_complex @ Y @ Y ) )
      = ( times_times_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_complex @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3928_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3929_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3930_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_3931_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3932_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3933_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_3934_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3935_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3936_mult__diff__mult,axiom,
    ! [X: complex,Y: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ Y ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3937_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_3938_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_3939_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_3940_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_3941_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_3942_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_3943_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_3944_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_3945_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_3946_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_3947_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_3948_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_3949_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_3950_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_3951_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_3952_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_3953_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3954_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_3955_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_3956_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_3957_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N2 @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_3958_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
      = ( ord_max_nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_3959_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3960_finite__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 )
              & ( ( size_s3941691890525107288d_enat @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3961_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3962_finite__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_int @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3963_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_nat @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3964_finite__lists__length__le,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3965_finite__lists__length__le,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3966_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3967_finite__lists__length__le,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3968_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3969_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3970_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_3971_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3972_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_3973_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3974_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_3975_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3976_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_3977_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3978_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3979_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3980_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3981_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3982_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3983_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3984_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3985_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_3986_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_3987_square__diff__one__factored,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_3988_inf__period_I3_J,axiom,
    ! [D: int,D6: int,T: int] :
      ( ( dvd_dvd_int @ D @ D6 )
     => ! [X3: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3989_inf__period_I3_J,axiom,
    ! [D: real,D6: real,T: real] :
      ( ( dvd_dvd_real @ D @ D6 )
     => ! [X3: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3990_inf__period_I3_J,axiom,
    ! [D: complex,D6: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D6 )
     => ! [X3: complex,K4: complex] :
          ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X3 @ T ) )
          = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3991_inf__period_I4_J,axiom,
    ! [D: int,D6: int,T: int] :
      ( ( dvd_dvd_int @ D @ D6 )
     => ! [X3: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3992_inf__period_I4_J,axiom,
    ! [D: real,D6: real,T: real] :
      ( ( dvd_dvd_real @ D @ D6 )
     => ! [X3: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3993_inf__period_I4_J,axiom,
    ! [D: complex,D6: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D6 )
     => ! [X3: complex,K4: complex] :
          ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3994_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_3995_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_3996_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_3997_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_3998_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_3999_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4000_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4001_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4002_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_4003_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_4004_nat__diff__split__asm,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P2 @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D4 ) )
                & ~ ( P2 @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4005_nat__diff__split,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P2 @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D4 ) )
           => ( P2 @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_4006_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_4007_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_4008_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4009_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_4010_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4011_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_4012_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4013_dvd__minus__add,axiom,
    ! [Q2: nat,N2: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N2 )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4014_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N2 )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4015_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4016_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4017_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_4018_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_4019_power__diff,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4020_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4021_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4022_power__diff,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4023_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_4024_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_4025_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M2 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_4026_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_4027_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_4028_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_4029_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4030_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: 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 ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_4031_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_4032_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4033_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4034_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4035_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4036_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4037_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4038_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4039_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P6: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4040_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P6: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4041_power__eq__if,axiom,
    ( power_8040749407984259932d_enat
    = ( ^ [P6: extended_enat,M2: nat] : ( if_Extended_enat @ ( M2 = zero_zero_nat ) @ one_on7984719198319812577d_enat @ ( times_7803423173614009249d_enat @ P6 @ ( power_8040749407984259932d_enat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4042_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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_4043_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_4044_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_4045_power__minus__mult,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_4046_power__minus__mult,axiom,
    ! [N2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_4047_power__minus__mult,axiom,
    ! [N2: nat,A: extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_7803423173614009249d_enat @ ( power_8040749407984259932d_enat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_8040749407984259932d_enat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_4048_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_4049_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_4050_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_4051_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_4052_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_4053_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_4054_bits__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P2 @ A5 ) )
     => ( ! [A5: nat,B4: $o] :
            ( ( P2 @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P2 @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_4055_bits__induct,axiom,
    ! [P2: int > $o,A: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P2 @ A5 ) )
     => ( ! [A5: int,B4: $o] :
            ( ( P2 @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P2 @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_4056_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_4057_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_4058_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_4059_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4060_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4061_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_4062_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_4063_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_4064_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4065_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4066_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_4067_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B4 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_4068_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B4 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_4069_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_4070_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_4071_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4072_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4073_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4074_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4075_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B4 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_4076_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_4077_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_4078_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_4079_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B4 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_4080_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ 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 @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4081_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ 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 @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4082_inrange,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_4083_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_4084_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z6: real] :
              ( ( power_power_real @ Z6 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_4085_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_4086_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B4 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4087_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4088_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4089_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_o,X: $o > nat,Y: $o > nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I5: $o] :
              ( ( member_o2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I5: $o] :
                ( ( member_o2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I5: $o] :
                ( ( member_o2 @ I5 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4090_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > nat,Y: real > nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real2 @ I5 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4091_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat2 @ I5 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4092_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > nat,Y: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_nat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_nat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4093_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > nat,Y: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int2 @ I5 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4094_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_Extended_enat,X: extended_enat > nat,Y: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I5: extended_enat] :
              ( ( member_Extended_enat @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_nat ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I5: extended_enat] :
                ( ( member_Extended_enat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_nat ) ) ) )
       => ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I5: extended_enat] :
                ( ( member_Extended_enat @ I5 @ I6 )
                & ( ( plus_plus_nat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4095_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_o,X: $o > real,Y: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I5: $o] :
              ( ( member_o2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I5: $o] :
                ( ( member_o2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I5: $o] :
                ( ( member_o2 @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4096_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real2 @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4097_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat2 @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat2 @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat2 @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4098_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4099_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_eq_int @ A @ I5 )
            & ( ord_less_eq_int @ I5 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_4100_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_int @ A @ I5 )
            & ( ord_less_int @ I5 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_4101_Icc__eq__Icc,axiom,
    ! [L: filter_nat,H2: filter_nat,L3: filter_nat,H3: filter_nat] :
      ( ( ( set_or1955772592623580779er_nat @ L @ H2 )
        = ( set_or1955772592623580779er_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_le2510731241096832064er_nat @ L @ H2 )
          & ~ ( ord_le2510731241096832064er_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4102_Icc__eq__Icc,axiom,
    ! [L: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H2 )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H2 )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4103_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_4104_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_4105_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_4106_atLeastAtMost__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o2 @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_eq_o @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4107_atLeastAtMost__iff,axiom,
    ! [I: filter_nat,L: filter_nat,U: filter_nat] :
      ( ( member_filter_nat @ I @ ( set_or1955772592623580779er_nat @ L @ U ) )
      = ( ( ord_le2510731241096832064er_nat @ L @ I )
        & ( ord_le2510731241096832064er_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4108_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat2 @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4109_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat2 @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4110_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int2 @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4111_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real2 @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4112_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_4113_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_eq_int @ A @ I5 )
            & ( ord_less_int @ I5 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_4114_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_int @ A @ I5 )
            & ( ord_less_eq_int @ I5 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_4115_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A @ B ) )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4116_atLeastatMost__empty__iff2,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( bot_bo498966703094740906er_nat
        = ( set_or1955772592623580779er_nat @ A @ B ) )
      = ( ~ ( ord_le2510731241096832064er_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4117_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4118_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4119_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4120_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4121_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4122_atLeastatMost__empty__iff,axiom,
    ! [A: filter_nat,B: filter_nat] :
      ( ( ( set_or1955772592623580779er_nat @ A @ B )
        = bot_bo498966703094740906er_nat )
      = ( ~ ( ord_le2510731241096832064er_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4123_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4124_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4125_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4126_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4127_atLeastatMost__subset__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat,D: filter_nat] :
      ( ( ord_le2426478655948331894er_nat @ ( set_or1955772592623580779er_nat @ A @ B ) @ ( set_or1955772592623580779er_nat @ C @ D ) )
      = ( ~ ( ord_le2510731241096832064er_nat @ A @ B )
        | ( ( ord_le2510731241096832064er_nat @ C @ A )
          & ( ord_le2510731241096832064er_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4128_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4129_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4130_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4131_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4132_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_o @ B @ A )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_4133_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( set_or5403411693681687835d_enat @ A @ B )
        = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
thf(fact_4134_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_4135_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_4136_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_4137_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_4138_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_4139_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_4140_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ( dvd_dvd_int @ M @ N2 )
         => ( ( dvd_dvd_int @ N2 @ M )
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_4141_zdvd__imp__le,axiom,
    ! [Z: int,N2: int] :
      ( ( dvd_dvd_int @ Z @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_4142_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_4143_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_4144_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N2 )
       => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).

% zdvd_not_zless
thf(fact_4145_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_4146_int__induct,axiom,
    ! [P2: int > $o,K: int,I: int] :
      ( ( P2 @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P2 @ I3 )
             => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_induct
thf(fact_4147_int__ge__induct,axiom,
    ! [K: int,I: int,P2: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P2 @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_ge_induct
thf(fact_4148_int__le__induct,axiom,
    ! [I: int,K: int,P2: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P2 @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_le_induct
thf(fact_4149_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_4150_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_4151_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_4152_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(3)
thf(fact_4153_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_4154_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_4155_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_4156_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_4157_int__less__induct,axiom,
    ! [I: int,K: int,P2: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_less_induct
thf(fact_4158_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_4159_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_4160_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_4161_int__gr__induct,axiom,
    ! [K: int,I: int,P2: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% int_gr_induct
thf(fact_4162_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_4163_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_4164_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_4165_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_4166_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(1)
thf(fact_4167_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_4168_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_4169_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_4170_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_4171_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_4172_all__nat__less,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
           => ( P2 @ M2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P2 @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_4173_ex__nat__less,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
            & ( P2 @ M2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P2 @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_4174_atLeastatMost__psubset__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ ( set_or5403411693681687835d_enat @ A @ B ) @ ( set_or5403411693681687835d_enat @ C @ D ) )
      = ( ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
          | ( ( ord_le2932123472753598470d_enat @ C @ A )
            & ( ord_le2932123472753598470d_enat @ B @ D )
            & ( ( ord_le72135733267957522d_enat @ C @ A )
              | ( ord_le72135733267957522d_enat @ B @ D ) ) ) )
        & ( ord_le2932123472753598470d_enat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4175_atLeastatMost__psubset__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat,D: filter_nat] :
      ( ( ord_le6505334834405097322er_nat @ ( set_or1955772592623580779er_nat @ A @ B ) @ ( set_or1955772592623580779er_nat @ C @ D ) )
      = ( ( ~ ( ord_le2510731241096832064er_nat @ A @ B )
          | ( ( ord_le2510731241096832064er_nat @ C @ A )
            & ( ord_le2510731241096832064er_nat @ B @ D )
            & ( ( ord_less_filter_nat @ C @ A )
              | ( ord_less_filter_nat @ B @ D ) ) ) )
        & ( ord_le2510731241096832064er_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4176_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4177_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4178_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4179_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4180_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_4181_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( 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] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( 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] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_4182_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B4 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( 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] :
                  ( ( X
                    = ( 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] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_4183_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B4 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_4184_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_4185_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_4186_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B4 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4187_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4188_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4189_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( 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] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4190_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_4191_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_4192_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_4193_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4194_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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_4195_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_4196_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_4197_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_4198_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_4199_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_4200_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4201_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4202_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4203_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4204_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_4205_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) )
      = ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4206_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_4207_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4208_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4209_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4210_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4211_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4212_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4213_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4214_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4215_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4216_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_4217_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_4218_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4219_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4220_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4221_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4222_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4223_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4224_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4225_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4226_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4227_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4228_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4229_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4230_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4231_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4232_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4233_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4234_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4235_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4236_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4237_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_4238_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_4239_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4240_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4241_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4242_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4243_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4244_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4245_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_4246_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_4247_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4248_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4249_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_4250_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_4251_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4252_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4253_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_4254_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_4255_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_4256_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_4257_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_4258_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_4259_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4260_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4261_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4262_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4263_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4264_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4265_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4266_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4267_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4268_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4269_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4270_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4271_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4272_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4273_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4274_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4275_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4276_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4277_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4278_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4279_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_4280_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_4281_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_4282_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_4283_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_4284_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_4285_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_4286_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_4287_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_4288_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_4289_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_4290_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_4291_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_4292_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_4293_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_4294_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_4295_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_4296_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_4297_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_4298_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4299_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4300_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4301_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4302_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4303_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4304_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_4305_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_4306_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4307_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4308_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4309_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4310_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4311_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4312_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_4313_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_4314_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4315_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4316_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4317_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4318_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_4319_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_4320_signed__take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_4321_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4322_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4323_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_4324_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_4325_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_4326_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_4327_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_4328_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_4329_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_4330_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_4331_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_4332_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_4333_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_4334_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_4335_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_4336_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_4337_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_4338_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_4339_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_4340_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_4341_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_4342_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_4343_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_4344_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_4345_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_4346_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4347_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_4348_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4349_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_4350_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4351_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_4352_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_4353_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_4354_minus__divide__left,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_4355_minus__divide__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( divide_divide_real @ A @ B ) ) ).

% minus_divide_divide
thf(fact_4356_minus__divide__right,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_divide_right
thf(fact_4357_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_4358_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_4359_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_4360_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_4361_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_4362_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_4363_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_4364_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_4365_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_4366_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_4367_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_4368_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_4369_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_4370_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_4371_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_4372_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_4373_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_4374_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_4375_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_4376_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4377_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4378_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_4379_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4380_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4381_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_4382_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4383_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4384_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_4385_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4386_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4387_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_4388_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4389_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4390_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_4391_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_4392_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_4393_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_4394_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_4395_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_4396_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_4397_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_4398_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_4399_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_4400_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_4401_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_4402_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4403_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4404_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4405_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4406_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4407_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_4408_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_4409_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_4410_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_4411_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_4412_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_4413_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_4414_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_4415_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_4416_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_4417_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_4418_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_4419_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_4420_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_4421_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_4422_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_4423_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_4424_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_4425_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_4426_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_4427_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_4428_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_4429_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_4430_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_4431_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_4432_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_4433_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_4434_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_4435_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_4436_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_4437_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_4438_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_4439_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_4440_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_4441_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_4442_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_4443_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_4444_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_4445_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_4446_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_4447_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_4448_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_4449_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_4450_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_4451_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_4452_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_4453_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_4454_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_4455_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4456_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_4457_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4458_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_4459_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4460_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_4461_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4462_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_4463_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4464_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_4465_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_4466_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_4467_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_4468_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_4469_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_4470_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_4471_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4472_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_4473_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_4474_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( 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_4475_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4476_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_4477_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4478_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_4479_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4480_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_4481_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4482_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_4483_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4484_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_4485_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_4486_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_4487_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_4488_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_4489_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_4490_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_4491_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ 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_4492_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( 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 ) @ B ) )
            & ( ~ ( 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_4493_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4494_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4495_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4496_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ 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_4497_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( 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 ) @ B ) )
            & ( ~ ( 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_4498_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_4499_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_4500_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_4501_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_4502_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_4503_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_4504_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_4505_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4506_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4507_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
        = ( ~ ( member_nat2 @ zero_zero_nat @ A2 ) ) ) ) ).

% even_set_encode_iff
thf(fact_4508_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_4509_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A3: nat] : ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4510_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4511_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4512_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4513_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4514_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4515_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_4516_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_4517_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4518_case__prod__conv,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4519_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4520_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4521_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4522_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_4523_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_4524_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_4525_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_4526_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_4527_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4528_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4529_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_4530_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_4531_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_4532_even__take__bit__eq,axiom,
    ! [N2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4533_even__take__bit__eq,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_4534_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_4535_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_4536_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_4537_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_4538_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4539_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4540_take__bit__Suc__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4541_take__bit__Suc__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_4542_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R4: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_4543_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R4: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_4544_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_4545_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_4546_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4547_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4548_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R4: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R4 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_4549_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R4: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R4 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_4550_take__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_4551_take__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_4552_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = ( bit_se2925701944663578781it_nat @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_4553_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_4554_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_4555_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_4556_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q2 )
         => ( ( F @ X5 @ Y3 )
            = ( G @ X5 @ Y3 ) ) )
     => ( ( P4 = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F @ P4 )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_4557_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > nat,G: nat > nat > nat,P4: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q2 )
         => ( ( F @ X5 @ Y3 )
            = ( G @ X5 @ Y3 ) ) )
     => ( ( P4 = Q2 )
       => ( ( produc6842872674320459806at_nat @ F @ P4 )
          = ( produc6842872674320459806at_nat @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_4558_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P4: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q2 )
         => ( ( F @ X5 @ Y3 )
            = ( G @ X5 @ Y3 ) ) )
     => ( ( P4 = Q2 )
       => ( ( produc27273713700761075at_nat @ F @ P4 )
          = ( produc27273713700761075at_nat @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_4559_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X5 @ Y3 )
            = Q2 )
         => ( ( F @ X5 @ Y3 )
            = ( G @ X5 @ Y3 ) ) )
     => ( ( P4 = Q2 )
       => ( ( produc8739625826339149834_nat_o @ F @ P4 )
          = ( produc8739625826339149834_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_4560_split__cong,axiom,
    ! [Q2: product_prod_int_int,F: int > int > $o,G: int > int > $o,P4: product_prod_int_int] :
      ( ! [X5: int,Y3: int] :
          ( ( ( product_Pair_int_int @ X5 @ Y3 )
            = Q2 )
         => ( ( F @ X5 @ Y3 )
            = ( G @ X5 @ Y3 ) ) )
     => ( ( P4 = Q2 )
       => ( ( produc4947309494688390418_int_o @ F @ P4 )
          = ( produc4947309494688390418_int_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_4561_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4562_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > nat,X1: nat,X22: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4563_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4564_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4565_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X22: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_4566_case__prodE2,axiom,
    ! [Q: $o > $o,P2: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P2 @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4567_case__prodE2,axiom,
    ! [Q: nat > $o,P2: nat > nat > nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6842872674320459806at_nat @ P2 @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4568_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P2 @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4569_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P2 @ Z ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4570_case__prodE2,axiom,
    ! [Q: $o > $o,P2: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P2 @ Z ) )
     => ~ ! [X5: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X5 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X5 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_4571_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4572_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat
        @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4573_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4574_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4575_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_4576_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F @ X5 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4577_cond__case__prod__eta,axiom,
    ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F @ X5 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc6842872674320459806at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4578_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F @ X5 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4579_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X5: nat,Y3: nat] :
          ( ( F @ X5 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4580_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X5: int,Y3: int] :
          ( ( F @ X5 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X5 @ Y3 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_4581_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_4582_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ A )
        = ( bit_ri631733984087533419it_int @ N2 @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_4583_signed__take__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_4584_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_4585_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_4586_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_4587_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_4588_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_4589_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_4590_take__bit__signed__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_4591_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_4592_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_4593_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_Bit1
thf(fact_4594_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_4595_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_4596_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_4597_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_4598_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_4599_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_4600_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_4601_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_code(3)
thf(fact_4602_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_4603_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_4604_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_4605_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_4606_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_4607_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_4608_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_4609_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4610_take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4611_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_4612_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_4613_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_4614_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_4615_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_4616_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_4617_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4618_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4619_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4620_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4621_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_4622_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_4623_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4624_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_4625_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_4626_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4627_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4628_take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4629_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_4630_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_4631_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4632_stable__imp__take__bit__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4633_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R4 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_4634_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( 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 @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_4635_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_4636_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_4637_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M2 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_4638_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K2: int] :
          ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4639_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K2: int] :
          ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4640_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K2: int] :
          ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_4641_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_4642_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_4643_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_4644_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4645_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4646_of__int__eq__iff,axiom,
    ! [W2: int,Z: int] :
      ( ( ( ring_1_of_int_real @ W2 )
        = ( ring_1_of_int_real @ Z ) )
      = ( W2 = Z ) ) ).

% of_int_eq_iff
thf(fact_4647_case__prodI2,axiom,
    ! [P4: produc859450856879609959at_nat,C: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P4
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc6590410687421337004_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_4648_case__prodI2,axiom,
    ! [P4: produc9072475918466114483BT_nat,C: vEBT_VEBT > nat > $o] :
      ( ! [A5: vEBT_VEBT,B4: nat] :
          ( ( P4
            = ( produc738532404422230701BT_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc7574032145190910526_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_4649_case__prodI2,axiom,
    ! [P4: produc7272778201969148633d_enat,C: vEBT_VEBT > extended_enat > $o] :
      ( ! [A5: vEBT_VEBT,B4: extended_enat] :
          ( ( P4
            = ( produc581526299967858633d_enat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc1206667027950169146enat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_4650_case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A5: nat,B4: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_4651_case__prodI2,axiom,
    ! [P4: product_prod_int_int,C: int > int > $o] :
      ( ! [A5: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_4652_case__prodI,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( F @ A @ B )
     => ( produc6590410687421337004_nat_o @ F @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_4653_case__prodI,axiom,
    ! [F: vEBT_VEBT > nat > $o,A: vEBT_VEBT,B: nat] :
      ( ( F @ A @ B )
     => ( produc7574032145190910526_nat_o @ F @ ( produc738532404422230701BT_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_4654_case__prodI,axiom,
    ! [F: vEBT_VEBT > extended_enat > $o,A: vEBT_VEBT,B: extended_enat] :
      ( ( F @ A @ B )
     => ( produc1206667027950169146enat_o @ F @ ( produc581526299967858633d_enat @ A @ B ) ) ) ).

% case_prodI
thf(fact_4655_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_4656_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( F @ A @ B )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_4657_mem__case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,Z: real,C: nat > nat > set_real] :
      ( ! [A5: nat,B4: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member_real2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_real2 @ Z @ ( produc3668448655016342576t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4658_mem__case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,Z: $o,C: nat > nat > set_o] :
      ( ! [A5: nat,B4: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member_o2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_o2 @ Z @ ( produc59986286002894506_set_o @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4659_mem__case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member_nat2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_nat2 @ Z @ ( produc6189476227299908564et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4660_mem__case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,Z: int,C: nat > nat > set_int] :
      ( ! [A5: nat,B4: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member_int2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_int2 @ Z @ ( produc2011625207790711856et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4661_mem__case__prodI2,axiom,
    ! [P4: produc9072475918466114483BT_nat,Z: real,C: vEBT_VEBT > nat > set_real] :
      ( ! [A5: vEBT_VEBT,B4: nat] :
          ( ( P4
            = ( produc738532404422230701BT_nat @ A5 @ B4 ) )
         => ( member_real2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_real2 @ Z @ ( produc4370036051912753340t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4662_mem__case__prodI2,axiom,
    ! [P4: produc9072475918466114483BT_nat,Z: $o,C: vEBT_VEBT > nat > set_o] :
      ( ! [A5: vEBT_VEBT,B4: nat] :
          ( ( P4
            = ( produc738532404422230701BT_nat @ A5 @ B4 ) )
         => ( member_o2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_o2 @ Z @ ( produc162892877563308318_set_o @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4663_mem__case__prodI2,axiom,
    ! [P4: produc9072475918466114483BT_nat,Z: nat,C: vEBT_VEBT > nat > set_nat] :
      ( ! [A5: vEBT_VEBT,B4: nat] :
          ( ( P4
            = ( produc738532404422230701BT_nat @ A5 @ B4 ) )
         => ( member_nat2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_nat2 @ Z @ ( produc8779078400790687328et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4664_mem__case__prodI2,axiom,
    ! [P4: produc9072475918466114483BT_nat,Z: int,C: vEBT_VEBT > nat > set_int] :
      ( ! [A5: vEBT_VEBT,B4: nat] :
          ( ( P4
            = ( produc738532404422230701BT_nat @ A5 @ B4 ) )
         => ( member_int2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_int2 @ Z @ ( produc4601227381281490620et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4665_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z: real,C: int > int > set_real] :
      ( ! [A5: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A5 @ B4 ) )
         => ( member_real2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_real2 @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4666_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z: $o,C: int > int > set_o] :
      ( ! [A5: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A5 @ B4 ) )
         => ( member_o2 @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member_o2 @ Z @ ( produc4257766111578684402_set_o @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_4667_mem__case__prodI,axiom,
    ! [Z: real,C: nat > nat > set_real,A: nat,B: nat] :
      ( ( member_real2 @ Z @ ( C @ A @ B ) )
     => ( member_real2 @ Z @ ( produc3668448655016342576t_real @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4668_mem__case__prodI,axiom,
    ! [Z: $o,C: nat > nat > set_o,A: nat,B: nat] :
      ( ( member_o2 @ Z @ ( C @ A @ B ) )
     => ( member_o2 @ Z @ ( produc59986286002894506_set_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4669_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat2 @ Z @ ( C @ A @ B ) )
     => ( member_nat2 @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4670_mem__case__prodI,axiom,
    ! [Z: int,C: nat > nat > set_int,A: nat,B: nat] :
      ( ( member_int2 @ Z @ ( C @ A @ B ) )
     => ( member_int2 @ Z @ ( produc2011625207790711856et_int @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4671_mem__case__prodI,axiom,
    ! [Z: real,C: vEBT_VEBT > nat > set_real,A: vEBT_VEBT,B: nat] :
      ( ( member_real2 @ Z @ ( C @ A @ B ) )
     => ( member_real2 @ Z @ ( produc4370036051912753340t_real @ C @ ( produc738532404422230701BT_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4672_mem__case__prodI,axiom,
    ! [Z: $o,C: vEBT_VEBT > nat > set_o,A: vEBT_VEBT,B: nat] :
      ( ( member_o2 @ Z @ ( C @ A @ B ) )
     => ( member_o2 @ Z @ ( produc162892877563308318_set_o @ C @ ( produc738532404422230701BT_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4673_mem__case__prodI,axiom,
    ! [Z: nat,C: vEBT_VEBT > nat > set_nat,A: vEBT_VEBT,B: nat] :
      ( ( member_nat2 @ Z @ ( C @ A @ B ) )
     => ( member_nat2 @ Z @ ( produc8779078400790687328et_nat @ C @ ( produc738532404422230701BT_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4674_mem__case__prodI,axiom,
    ! [Z: int,C: vEBT_VEBT > nat > set_int,A: vEBT_VEBT,B: nat] :
      ( ( member_int2 @ Z @ ( C @ A @ B ) )
     => ( member_int2 @ Z @ ( produc4601227381281490620et_int @ C @ ( produc738532404422230701BT_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4675_mem__case__prodI,axiom,
    ! [Z: real,C: int > int > set_real,A: int,B: int] :
      ( ( member_real2 @ Z @ ( C @ A @ B ) )
     => ( member_real2 @ Z @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4676_mem__case__prodI,axiom,
    ! [Z: $o,C: int > int > set_o,A: int,B: int] :
      ( ( member_o2 @ Z @ ( C @ A @ B ) )
     => ( member_o2 @ Z @ ( produc4257766111578684402_set_o @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_4677_case__prodI2_H,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( ( product_Pair_nat_nat @ A5 @ B4 )
            = P4 )
         => ( C @ A5 @ B4 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).

% case_prodI2'
thf(fact_4678_of__int__of__bool,axiom,
    ! [P2: $o] :
      ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( zero_n3304061248610475627l_real @ P2 ) ) ).

% of_int_of_bool
thf(fact_4679_of__int__of__bool,axiom,
    ! [P2: $o] :
      ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% of_int_of_bool
thf(fact_4680_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_4681_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_4682_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_4683_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_4684_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_4685_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4686_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_4687_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_4688_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4689_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_4690_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_4691_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_4692_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4693_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4694_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_4695_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_4696_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_4697_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_4698_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4699_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = one_one_complex )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4700_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_4701_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4702_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_4703_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_4704_of__int__mult,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_4705_of__int__mult,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_4706_of__int__mult,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_mult
thf(fact_4707_of__int__add,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_4708_of__int__add,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_4709_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_minus
thf(fact_4710_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_minus
thf(fact_4711_of__int__diff,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_4712_of__int__diff,axiom,
    ! [W2: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_4713_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_4714_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_4715_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_4716_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( X
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4717_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( X
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4718_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
      = ( X
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4719_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4720_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4721_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B @ W2 )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4722_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4723_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4724_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4725_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_4726_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_4727_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_4728_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_4729_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_4730_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% diff_numeral_Suc
thf(fact_4731_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_4732_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% max_numeral_Suc
thf(fact_4733_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_4734_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_4735_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_4736_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_4737_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_4738_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_4739_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_4740_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_4741_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4742_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4743_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4744_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4745_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4746_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4747_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4748_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4749_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_4750_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_4751_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_4752_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_4753_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_4754_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_4755_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_4756_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_4757_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4758_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4759_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4760_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4761_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4762_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4763_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4764_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4765_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4766_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4767_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4768_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4769_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_4770_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4771_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4772_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_4773_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4774_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4775_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4776_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4777_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4778_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4779_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4780_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4781_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4782_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4783_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4784_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4785_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4786_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4787_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4788_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4789_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4790_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4791_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4792_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4793_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4794_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4795_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4796_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_4797_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_le_of_int
thf(fact_4798_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_less_of_int
thf(fact_4799_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_4800_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_4801_mult__of__int__commute,axiom,
    ! [X: int,Y: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).

% mult_of_int_commute
thf(fact_4802_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).

% mult_of_int_commute
thf(fact_4803_mem__case__prodE,axiom,
    ! [Z: real,C: nat > nat > set_real,P4: product_prod_nat_nat] :
      ( ( member_real2 @ Z @ ( produc3668448655016342576t_real @ C @ P4 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_real2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4804_mem__case__prodE,axiom,
    ! [Z: $o,C: nat > nat > set_o,P4: product_prod_nat_nat] :
      ( ( member_o2 @ Z @ ( produc59986286002894506_set_o @ C @ P4 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_o2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4805_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > nat > set_nat,P4: product_prod_nat_nat] :
      ( ( member_nat2 @ Z @ ( produc6189476227299908564et_nat @ C @ P4 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_nat2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4806_mem__case__prodE,axiom,
    ! [Z: int,C: nat > nat > set_int,P4: product_prod_nat_nat] :
      ( ( member_int2 @ Z @ ( produc2011625207790711856et_int @ C @ P4 ) )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( member_int2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4807_mem__case__prodE,axiom,
    ! [Z: real,C: vEBT_VEBT > nat > set_real,P4: produc9072475918466114483BT_nat] :
      ( ( member_real2 @ Z @ ( produc4370036051912753340t_real @ C @ P4 ) )
     => ~ ! [X5: vEBT_VEBT,Y3: nat] :
            ( ( P4
              = ( produc738532404422230701BT_nat @ X5 @ Y3 ) )
           => ~ ( member_real2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4808_mem__case__prodE,axiom,
    ! [Z: $o,C: vEBT_VEBT > nat > set_o,P4: produc9072475918466114483BT_nat] :
      ( ( member_o2 @ Z @ ( produc162892877563308318_set_o @ C @ P4 ) )
     => ~ ! [X5: vEBT_VEBT,Y3: nat] :
            ( ( P4
              = ( produc738532404422230701BT_nat @ X5 @ Y3 ) )
           => ~ ( member_o2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4809_mem__case__prodE,axiom,
    ! [Z: nat,C: vEBT_VEBT > nat > set_nat,P4: produc9072475918466114483BT_nat] :
      ( ( member_nat2 @ Z @ ( produc8779078400790687328et_nat @ C @ P4 ) )
     => ~ ! [X5: vEBT_VEBT,Y3: nat] :
            ( ( P4
              = ( produc738532404422230701BT_nat @ X5 @ Y3 ) )
           => ~ ( member_nat2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4810_mem__case__prodE,axiom,
    ! [Z: int,C: vEBT_VEBT > nat > set_int,P4: produc9072475918466114483BT_nat] :
      ( ( member_int2 @ Z @ ( produc4601227381281490620et_int @ C @ P4 ) )
     => ~ ! [X5: vEBT_VEBT,Y3: nat] :
            ( ( P4
              = ( produc738532404422230701BT_nat @ X5 @ Y3 ) )
           => ~ ( member_int2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4811_mem__case__prodE,axiom,
    ! [Z: real,C: int > int > set_real,P4: product_prod_int_int] :
      ( ( member_real2 @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) )
     => ~ ! [X5: int,Y3: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y3 ) )
           => ~ ( member_real2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4812_mem__case__prodE,axiom,
    ! [Z: $o,C: int > int > set_o,P4: product_prod_int_int] :
      ( ( member_o2 @ Z @ ( produc4257766111578684402_set_o @ C @ P4 ) )
     => ~ ! [X5: int,Y3: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y3 ) )
           => ~ ( member_o2 @ Z @ ( C @ X5 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_4813_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_real @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).

% of_int_max
thf(fact_4814_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).

% of_int_max
thf(fact_4815_case__prodE,axiom,
    ! [C: product_prod_nat_nat > product_prod_nat_nat > $o,P4: produc859450856879609959at_nat] :
      ( ( produc6590410687421337004_nat_o @ C @ P4 )
     => ~ ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( P4
              = ( produc6161850002892822231at_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_4816_case__prodE,axiom,
    ! [C: vEBT_VEBT > nat > $o,P4: produc9072475918466114483BT_nat] :
      ( ( produc7574032145190910526_nat_o @ C @ P4 )
     => ~ ! [X5: vEBT_VEBT,Y3: nat] :
            ( ( P4
              = ( produc738532404422230701BT_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_4817_case__prodE,axiom,
    ! [C: vEBT_VEBT > extended_enat > $o,P4: produc7272778201969148633d_enat] :
      ( ( produc1206667027950169146enat_o @ C @ P4 )
     => ~ ! [X5: vEBT_VEBT,Y3: extended_enat] :
            ( ( P4
              = ( produc581526299967858633d_enat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_4818_case__prodE,axiom,
    ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P4 )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_4819_case__prodE,axiom,
    ! [C: int > int > $o,P4: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P4 )
     => ~ ! [X5: int,Y3: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 ) ) ) ).

% case_prodE
thf(fact_4820_case__prodD,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( produc6590410687421337004_nat_o @ F @ ( produc6161850002892822231at_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_4821_case__prodD,axiom,
    ! [F: vEBT_VEBT > nat > $o,A: vEBT_VEBT,B: nat] :
      ( ( produc7574032145190910526_nat_o @ F @ ( produc738532404422230701BT_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_4822_case__prodD,axiom,
    ! [F: vEBT_VEBT > extended_enat > $o,A: vEBT_VEBT,B: extended_enat] :
      ( ( produc1206667027950169146enat_o @ F @ ( produc581526299967858633d_enat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_4823_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_4824_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_4825_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
     => ~ ! [X5: nat,Y3: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y3 ) )
           => ~ ( C @ X5 @ Y3 @ Z ) ) ) ).

% case_prodE'
thf(fact_4826_case__prodD_H,axiom,
    ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_4827_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_4828_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_4829_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_4830_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_4831_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_4832_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_4833_floor__exists,axiom,
    ! [X: real] :
    ? [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_exists
thf(fact_4834_floor__exists1,axiom,
    ! [X: real] :
    ? [X5: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X5 ) ) ) ).

% floor_exists1
thf(fact_4835_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_4836_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_4837_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera1916890842035813515d_enat @ ( inc @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_inc
thf(fact_4838_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_4839_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_real @ ( inc @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_4840_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4841_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4842_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M2: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_4843_listrel1p__def,axiom,
    ( listrel1p_nat
    = ( ^ [R4: nat > nat > $o,Xs2: list_nat,Ys3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( listrel1_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R4 ) ) ) ) ) ) ).

% listrel1p_def
thf(fact_4844_listrel1p__def,axiom,
    ( listrel1p_int
    = ( ^ [R4: int > int > $o,Xs2: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R4 ) ) ) ) ) ) ).

% listrel1p_def
thf(fact_4845_round__unique,axiom,
    ! [X: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_4846_of__int__round__gt,axiom,
    ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_gt
thf(fact_4847_of__int__round__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_ge
thf(fact_4848_of__int__round__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_4849_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D4 @ Z6 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_4850_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D4 @ Z7 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_4851_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_4852_round__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim8280529875227126926d_real @ Y ) ) ) ).

% round_mono
thf(fact_4853_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_4854_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_4855_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_4856_set__encode__insert,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat2 @ N2 @ A2 )
       => ( ( nat_set_encode @ ( insert_nat2 @ N2 @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_4857_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_4858_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_4859_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_4860_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I5 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_4861_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_4862_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat2 @ A @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_4863_singletonI,axiom,
    ! [A: real] : ( member_real2 @ A @ ( insert_real2 @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_4864_singletonI,axiom,
    ! [A: $o] : ( member_o2 @ A @ ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_4865_singletonI,axiom,
    ! [A: nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_4866_singletonI,axiom,
    ! [A: int] : ( member_int2 @ A @ ( insert_int2 @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_4867_insert__subset,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real2 @ X @ A2 ) @ B2 )
      = ( ( member_real2 @ X @ B2 )
        & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4868_insert__subset,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o2 @ X @ A2 ) @ B2 )
      = ( ( member_o2 @ X @ B2 )
        & ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4869_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ X @ A2 ) @ B2 )
      = ( ( member_set_nat2 @ X @ B2 )
        & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4870_insert__subset,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int2 @ X @ A2 ) @ B2 )
      = ( ( member_int2 @ X @ B2 )
        & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4871_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A2 ) @ B2 )
      = ( ( member_nat2 @ X @ B2 )
        & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4872_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_4873_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_4874_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_4875_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_4876_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_4877_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X2: list_nat] : X2 = A )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_4878_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X2: set_nat] : X2 = A )
      = ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_4879_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X2: real] : X2 = A )
      = ( insert_real2 @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_4880_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X2: $o] : X2 = A )
      = ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_4881_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : X2 = A )
      = ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_4882_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X2: int] : X2 = A )
      = ( insert_int2 @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_4883_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y5: list_nat,Z3: list_nat] : Y5 = Z3
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_4884_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3
          @ A ) )
      = ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_4885_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y5: real,Z3: real] : Y5 = Z3
          @ A ) )
      = ( insert_real2 @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_4886_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y5: $o,Z3: $o] : Y5 = Z3
          @ A ) )
      = ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_4887_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z3: nat] : Y5 = Z3
          @ A ) )
      = ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_4888_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z3: int] : Y5 = Z3
          @ A ) )
      = ( insert_int2 @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_4889_of__int__sum,axiom,
    ! [F: complex > int,A2: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( ring_17405671764205052669omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_4890_of__int__sum,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_4891_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_4892_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_4893_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_4894_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_4895_sum_Oempty,axiom,
    ! [G: real > extended_enat] :
      ( ( groups2800946370649118462d_enat @ G @ bot_bot_set_real )
      = zero_z5237406670263579293d_enat ) ).

% sum.empty
thf(fact_4896_sum_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_4897_sum_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups8691415230153176458o_real @ G @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_4898_sum_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_4899_sum_Oempty,axiom,
    ! [G: $o > complex] :
      ( ( groups5328290441151304332omplex @ G @ bot_bot_set_o )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_4900_sum_Oempty,axiom,
    ! [G: $o > extended_enat] :
      ( ( groups7198740251461348360d_enat @ G @ bot_bot_set_o )
      = zero_z5237406670263579293d_enat ) ).

% sum.empty
thf(fact_4901_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_4902_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_4903_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_4904_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_4905_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_4906_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_4907_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_4908_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_4909_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > int] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4538972089207619220nt_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_4910_sum_Oinfinite,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2025484359314973016at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_4911_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X2: complex] :
              ( ( member_complex @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4912_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X2: int] :
              ( ( member_int2 @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4913_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ( groups2027974829824023292at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4914_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups7108830773950497114d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X2: nat] :
              ( ( member_nat2 @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4915_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups1752964319039525884d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X2: complex] :
              ( ( member_complex @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4916_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4225252721152677374d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X2: int] :
              ( ( member_int2 @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4917_sum__eq__0__iff,axiom,
    ! [F3: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ( groups2433450451889696826d_enat @ F @ F3 )
          = zero_z5237406670263579293d_enat )
        = ( ! [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4918_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X2: nat] :
              ( ( member_nat2 @ X2 @ F3 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_4919_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real2 @ A @ A2 )
        = ( insert_real2 @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4920_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( ( insert_o2 @ A @ A2 )
        = ( insert_o2 @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4921_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int2 @ A @ A2 )
        = ( insert_int2 @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4922_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat2 @ A @ A2 )
        = ( insert_nat2 @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4923_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real2 @ B @ bot_bot_set_real )
        = ( insert_real2 @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4924_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( ( insert_o2 @ B @ bot_bot_set_o )
        = ( insert_o2 @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4925_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int2 @ B @ bot_bot_set_int )
        = ( insert_int2 @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4926_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat2 @ B @ bot_bot_set_nat )
        = ( insert_nat2 @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4927_atLeastAtMost__singleton,axiom,
    ! [A: $o] :
      ( ( set_or8904488021354931149Most_o @ A @ A )
      = ( insert_o2 @ A @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_4928_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_4929_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int2 @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_4930_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real2 @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_4931_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o2 @ C @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4932_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat2 @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4933_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int2 @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4934_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real2 @ C @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4935_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real2 @ A @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
      = ( insert_real2 @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4936_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o2 @ A @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
      = ( insert_o2 @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4937_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int2 @ A @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
      = ( insert_int2 @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4938_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat2 @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4939_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4940_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_4941_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_4942_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N2 )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_4943_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N2 )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_4944_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_4945_sum_Odelta,axiom,
    ! [S3: set_o,A: $o,B: $o > nat] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o2 @ A @ S3 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K2: $o] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_o2 @ A @ S3 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K2: $o] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_4946_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_4947_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int2 @ A @ S3 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int2 @ A @ S3 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_4948_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_4949_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_4950_sum_Odelta,axiom,
    ! [S3: set_o,A: $o,B: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o2 @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K2: $o] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_o2 @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K2: $o] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_4951_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_4952_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int2 @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int2 @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_4953_sum_Odelta,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_4954_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_4955_sum_Odelta_H,axiom,
    ! [S3: set_o,A: $o,B: $o > nat] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o2 @ A @ S3 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K2: $o] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_o2 @ A @ S3 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K2: $o] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_4956_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_4957_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int2 @ A @ S3 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int2 @ A @ S3 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_4958_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S3 )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_4959_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_4960_sum_Odelta_H,axiom,
    ! [S3: set_o,A: $o,B: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o2 @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K2: $o] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_o2 @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K2: $o] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_4961_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_4962_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int2 @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int2 @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_4963_sum_Odelta_H,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_4964_List_Oset__insert,axiom,
    ! [X: real,Xs: list_real] :
      ( ( set_real2 @ ( insert_real @ X @ Xs ) )
      = ( insert_real2 @ X @ ( set_real2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_4965_List_Oset__insert,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( set_o2 @ ( insert_o @ X @ Xs ) )
      = ( insert_o2 @ X @ ( set_o2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_4966_List_Oset__insert,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( insert_VEBT_VEBT @ X @ Xs ) )
      = ( insert_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_4967_List_Oset__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( set_int2 @ ( insert_int @ X @ Xs ) )
      = ( insert_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_4968_List_Oset__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
      = ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% List.set_insert
thf(fact_4969_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real2 @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real2 @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4970_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o2 @ X @ A2 )
       => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o2 @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4971_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4972_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int2 @ X @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int2 @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4973_sum_Oinsert,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4974_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real2 @ X @ A2 )
       => ( ( groups1932886352136224148al_int @ G @ ( insert_real2 @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4975_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o2 @ X @ A2 )
       => ( ( groups8505340233167759370_o_int @ G @ ( insert_o2 @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4976_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat2 @ X @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ ( insert_nat2 @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4977_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4978_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int2 @ X @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ ( insert_int2 @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_4979_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_4980_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_4981_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_4982_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_4983_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_4984_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_4985_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat2 @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat2 @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4986_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real2 @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real2 @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4987_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o2 @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o2 @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4988_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int2 @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int2 @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4989_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat2 @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_4990_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_4991_ceiling__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_4992_zero__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_ceiling
thf(fact_4993_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_4994_ceiling__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_4995_one__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_le_ceiling
thf(fact_4996_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_4997_ceiling__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_4998_one__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ one_one_real @ X ) ) ).

% one_less_ceiling
thf(fact_4999_set__replicate,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
        = ( insert_VEBT_VEBT2 @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_5000_set__replicate,axiom,
    ! [N2: nat,X: real] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
        = ( insert_real2 @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_5001_set__replicate,axiom,
    ! [N2: nat,X: $o] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_o2 @ ( replicate_o @ N2 @ X ) )
        = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ).

% set_replicate
thf(fact_5002_set__replicate,axiom,
    ! [N2: nat,X: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
        = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_5003_set__replicate,axiom,
    ! [N2: nat,X: int] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
        = ( insert_int2 @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_5004_ceiling__add__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5005_ceiling__add__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_5006_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_5007_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_5008_bit__0,axiom,
    ! [A: int] :
      ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5009_bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_5010_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5011_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5012_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5013_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5014_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5015_ceiling__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_5016_zero__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).

% zero_le_ceiling
thf(fact_5017_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_5018_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_5019_bit__mod__2__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 )
      = ( ( N2 = zero_zero_nat )
        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5020_bit__mod__2__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
      = ( ( N2 = zero_zero_nat )
        & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_5021_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_5022_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_5023_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_5024_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_5025_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_5026_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat2 @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_5027_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real2 @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real2 @ X @ A2 ) )
            = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
        & ( ~ ( member_real2 @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real2 @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5028_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o2 @ X @ A2 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o2 @ X @ A2 ) )
            = ( groups8507830703676809646_o_nat @ G @ A2 ) ) )
        & ( ~ ( member_o2 @ X @ A2 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o2 @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5029_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5030_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int2 @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int2 @ X @ A2 ) )
            = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
        & ( ~ ( member_int2 @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int2 @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5031_sum_Oinsert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( groups2027974829824023292at_nat @ G @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5032_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real2 @ X @ A2 )
         => ( ( groups1932886352136224148al_int @ G @ ( insert_real2 @ X @ A2 ) )
            = ( groups1932886352136224148al_int @ G @ A2 ) ) )
        & ( ~ ( member_real2 @ X @ A2 )
         => ( ( groups1932886352136224148al_int @ G @ ( insert_real2 @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5033_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o2 @ X @ A2 )
         => ( ( groups8505340233167759370_o_int @ G @ ( insert_o2 @ X @ A2 ) )
            = ( groups8505340233167759370_o_int @ G @ A2 ) ) )
        & ( ~ ( member_o2 @ X @ A2 )
         => ( ( groups8505340233167759370_o_int @ G @ ( insert_o2 @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5034_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat2 @ X @ A2 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat2 @ X @ A2 ) )
            = ( groups3539618377306564664at_int @ G @ A2 ) ) )
        & ( ~ ( member_nat2 @ X @ A2 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat2 @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5035_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5690904116761175830ex_int @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5036_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int2 @ X @ A2 )
         => ( ( groups4538972089207619220nt_int @ G @ ( insert_int2 @ X @ A2 ) )
            = ( groups4538972089207619220nt_int @ G @ A2 ) ) )
        & ( ~ ( member_int2 @ X @ A2 )
         => ( ( groups4538972089207619220nt_int @ G @ ( insert_int2 @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5037_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_5038_sum_Oneutral,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_5039_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_5040_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > nat,A2: set_real] :
      ( ( ( groups1935376822645274424al_nat @ G @ A2 )
       != zero_zero_nat )
     => ~ ! [A5: real] :
            ( ( member_real2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5041_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > nat,A2: set_o] :
      ( ( ( groups8507830703676809646_o_nat @ G @ A2 )
       != zero_zero_nat )
     => ~ ! [A5: $o] :
            ( ( member_o2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5042_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > nat,A2: set_int] :
      ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
       != zero_zero_nat )
     => ~ ! [A5: int] :
            ( ( member_int2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5043_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: real] :
            ( ( member_real2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5044_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > real,A2: set_o] :
      ( ( ( groups8691415230153176458o_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: $o] :
            ( ( member_o2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5045_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: int] :
            ( ( member_int2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5046_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > int,A2: set_real] :
      ( ( ( groups1932886352136224148al_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A5: real] :
            ( ( member_real2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5047_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > int,A2: set_o] :
      ( ( ( groups8505340233167759370_o_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A5: $o] :
            ( ( member_o2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5048_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > int,A2: set_nat] :
      ( ( ( groups3539618377306564664at_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A5: nat] :
            ( ( member_nat2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5049_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > int,A2: set_int] :
      ( ( ( groups4538972089207619220nt_int @ G @ A2 )
       != zero_zero_int )
     => ~ ! [A5: int] :
            ( ( member_int2 @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5050_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N3 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N3 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_5051_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N3 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N3 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_5052_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat2 @ B @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_5053_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real2 @ B @ ( insert_real2 @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_5054_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o2 @ B @ ( insert_o2 @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_5055_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat2 @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_5056_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int2 @ B @ ( insert_int2 @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_5057_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat2 @ B @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_5058_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real2 @ B @ ( insert_real2 @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_5059_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o2 @ B @ ( insert_o2 @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_5060_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat2 @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_5061_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int2 @ B @ ( insert_int2 @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_5062_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real2 @ A @ ( insert_real2 @ B @ bot_bot_set_real ) )
        = ( insert_real2 @ C @ ( insert_real2 @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5063_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o2 @ A @ ( insert_o2 @ B @ bot_bot_set_o ) )
        = ( insert_o2 @ C @ ( insert_o2 @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5064_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
        = ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5065_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int2 @ A @ ( insert_int2 @ B @ bot_bot_set_int ) )
        = ( insert_int2 @ C @ ( insert_int2 @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_5066_insert__not__empty,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real2 @ A @ A2 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_5067_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o2 @ A @ A2 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_5068_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat2 @ A @ A2 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_5069_insert__not__empty,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int2 @ A @ A2 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_5070_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real2 @ A @ bot_bot_set_real )
        = ( insert_real2 @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_5071_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o2 @ A @ bot_bot_set_o )
        = ( insert_o2 @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_5072_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat2 @ A @ bot_bot_set_nat )
        = ( insert_nat2 @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_5073_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int2 @ A @ bot_bot_set_int )
        = ( insert_int2 @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_5074_insert__subsetI,axiom,
    ! [X: real,A2: set_real,X7: set_real] :
      ( ( member_real2 @ X @ A2 )
     => ( ( ord_less_eq_set_real @ X7 @ A2 )
       => ( ord_less_eq_set_real @ ( insert_real2 @ X @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_5075_insert__subsetI,axiom,
    ! [X: $o,A2: set_o,X7: set_o] :
      ( ( member_o2 @ X @ A2 )
     => ( ( ord_less_eq_set_o @ X7 @ A2 )
       => ( ord_less_eq_set_o @ ( insert_o2 @ X @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_5076_insert__subsetI,axiom,
    ! [X: set_nat,A2: set_set_nat,X7: set_set_nat] :
      ( ( member_set_nat2 @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ X7 @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat2 @ X @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_5077_insert__subsetI,axiom,
    ! [X: int,A2: set_int,X7: set_int] :
      ( ( member_int2 @ X @ A2 )
     => ( ( ord_less_eq_set_int @ X7 @ A2 )
       => ( ord_less_eq_set_int @ ( insert_int2 @ X @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_5078_insert__subsetI,axiom,
    ! [X: nat,A2: set_nat,X7: set_nat] :
      ( ( member_nat2 @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ X7 @ A2 )
       => ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_5079_insert__mono,axiom,
    ! [C4: set_int,D6: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D6 )
     => ( ord_less_eq_set_int @ ( insert_int2 @ A @ C4 ) @ ( insert_int2 @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_5080_insert__mono,axiom,
    ! [C4: set_real,D6: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C4 @ D6 )
     => ( ord_less_eq_set_real @ ( insert_real2 @ A @ C4 ) @ ( insert_real2 @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_5081_insert__mono,axiom,
    ! [C4: set_o,D6: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C4 @ D6 )
     => ( ord_less_eq_set_o @ ( insert_o2 @ A @ C4 ) @ ( insert_o2 @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_5082_insert__mono,axiom,
    ! [C4: set_nat,D6: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C4 @ D6 )
     => ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C4 ) @ ( insert_nat2 @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_5083_subset__insert,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real2 @ X @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ X @ B2 ) )
        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_5084_subset__insert,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o2 @ X @ A2 )
     => ( ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ X @ B2 ) )
        = ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_5085_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat2 @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat2 @ X @ B2 ) )
        = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_5086_subset__insert,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int2 @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ X @ B2 ) )
        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_5087_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat2 @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_5088_subset__insertI,axiom,
    ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int2 @ A @ B2 ) ) ).

% subset_insertI
thf(fact_5089_subset__insertI,axiom,
    ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real2 @ A @ B2 ) ) ).

% subset_insertI
thf(fact_5090_subset__insertI,axiom,
    ! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o2 @ A @ B2 ) ) ).

% subset_insertI
thf(fact_5091_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat2 @ A @ B2 ) ) ).

% subset_insertI
thf(fact_5092_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_5093_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_5094_subset__insertI2,axiom,
    ! [A2: set_o,B2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_5095_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_5096_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_5097_sum__mono,axiom,
    ! [K5: set_real,F: real > real,G: real > real] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ K5 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K5 ) @ ( groups8097168146408367636l_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5098_sum__mono,axiom,
    ! [K5: set_o,F: $o > real,G: $o > real] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ K5 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ K5 ) @ ( groups8691415230153176458o_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5099_sum__mono,axiom,
    ! [K5: set_int,F: int > real,G: int > real] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ K5 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ K5 ) @ ( groups8778361861064173332t_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5100_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5101_sum__mono,axiom,
    ! [K5: set_o,F: $o > nat,G: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K5 ) @ ( groups8507830703676809646_o_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5102_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5103_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5104_sum__mono,axiom,
    ! [K5: set_o,F: $o > int,G: $o > int] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K5 ) @ ( groups8505340233167759370_o_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5105_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat2 @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5106_sum__mono,axiom,
    ! [K5: set_int,F: int > int,G: int > int] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_5107_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5108_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5109_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5110_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat2 @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_set_nat2 @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5111_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real2 @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real2 @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5112_sum__diff1__nat,axiom,
    ! [A: $o,A2: set_o,F: $o > nat] :
      ( ( ( member_o2 @ A @ A2 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_o2 @ A @ A2 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
          = ( groups8507830703676809646_o_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5113_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int2 @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int2 @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5114_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat2 @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat2 @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_5115_Collect__conv__if,axiom,
    ! [P2: list_nat > $o,A: list_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_5116_Collect__conv__if,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_5117_Collect__conv__if,axiom,
    ! [P2: real > $o,A: real] :
      ( ( ( P2 @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert_real2 @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_5118_Collect__conv__if,axiom,
    ! [P2: $o > $o,A: $o] :
      ( ( ( P2 @ A )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert_o2 @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_5119_Collect__conv__if,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_5120_Collect__conv__if,axiom,
    ! [P2: int > $o,A: int] :
      ( ( ( P2 @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert_int2 @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_5121_Collect__conv__if2,axiom,
    ! [P2: list_nat > $o,A: list_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_5122_Collect__conv__if2,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ( ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_5123_Collect__conv__if2,axiom,
    ! [P2: real > $o,A: real] :
      ( ( ( P2 @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert_real2 @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_5124_Collect__conv__if2,axiom,
    ! [P2: $o > $o,A: $o] :
      ( ( ( P2 @ A )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert_o2 @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_5125_Collect__conv__if2,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_5126_Collect__conv__if2,axiom,
    ! [P2: int > $o,A: int] :
      ( ( ( P2 @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert_int2 @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_5127_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5128_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2027974829824023292at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5129_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5130_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2025484359314973016at_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups2025484359314973016at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5131_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5132_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups4148127829035722712t_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5133_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups1752964319039525884d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups1752964319039525884d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5134_sum_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2433450451889696826d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups2433450451889696826d_enat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5135_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real2 @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5136_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real2 @ X @ A2 )
       => ( ( groups1932886352136224148al_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_5137_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5138_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5139_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5140_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2025484359314973016at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups2025484359314973016at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5141_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5142_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5143_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > extended_enat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups1752964319039525884d_enat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups1752964319039525884d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5144_sum_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2433450451889696826d_enat @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ X ) @ ( groups2433450451889696826d_enat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5145_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G @ ( insert_real2 @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5146_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > int,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G @ ( insert_real2 @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_5147_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5148_sum__diff1,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ A @ A2 )
         => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ A2 )
         => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups2025484359314973016at_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5149_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real2 @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real2 @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
            = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5150_sum__diff1,axiom,
    ! [A2: set_o,A: $o,F: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o2 @ A @ A2 )
         => ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_int @ ( groups8505340233167759370_o_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o2 @ A @ A2 )
         => ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
            = ( groups8505340233167759370_o_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5151_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int2 @ A @ A2 )
         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int2 @ A @ A2 )
         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
            = ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5152_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5153_sum__diff1,axiom,
    ! [A2: set_Extended_enat,A: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ A @ A2 )
         => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ A2 )
         => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups4148127829035722712t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5154_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real2 @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real2 @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5155_sum__diff1,axiom,
    ! [A2: set_o,A: $o,F: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o2 @ A @ A2 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o2 @ A @ A2 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
            = ( groups8691415230153176458o_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5156_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int2 @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int2 @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_5157_sum__nonneg,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5158_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > extended_enat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups7198740251461348360d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5159_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5160_sum__nonneg,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
     => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( groups4225252721152677374d_enat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5161_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5162_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5163_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5164_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5165_sum__nonneg,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5166_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_5167_sum__nonpos,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5168_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > extended_enat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7198740251461348360d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5169_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5170_sum__nonpos,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ zero_z5237406670263579293d_enat ) )
     => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ A2 ) @ zero_z5237406670263579293d_enat ) ) ).

% sum_nonpos
thf(fact_5171_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5172_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5173_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5174_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5175_sum__nonpos,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5176_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5177_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_5178_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_5179_sum__mono__inv,axiom,
    ! [F: real > real,I6: set_real,G: real > real,I: real] :
      ( ( ( groups8097168146408367636l_real @ F @ I6 )
        = ( groups8097168146408367636l_real @ G @ I6 ) )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ I6 )
           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real2 @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5180_sum__mono__inv,axiom,
    ! [F: $o > real,I6: set_o,G: $o > real,I: $o] :
      ( ( ( groups8691415230153176458o_real @ F @ I6 )
        = ( groups8691415230153176458o_real @ G @ I6 ) )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ I6 )
           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o2 @ I @ I6 )
         => ( ( finite_finite_o @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5181_sum__mono__inv,axiom,
    ! [F: complex > real,I6: set_complex,G: complex > real,I: complex] :
      ( ( ( groups5808333547571424918x_real @ F @ I6 )
        = ( groups5808333547571424918x_real @ G @ I6 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I6 )
           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5182_sum__mono__inv,axiom,
    ! [F: int > real,I6: set_int,G: int > real,I: int] :
      ( ( ( groups8778361861064173332t_real @ F @ I6 )
        = ( groups8778361861064173332t_real @ G @ I6 ) )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ I6 )
           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int2 @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5183_sum__mono__inv,axiom,
    ! [F: extended_enat > real,I6: set_Extended_enat,G: extended_enat > real,I: extended_enat] :
      ( ( ( groups4148127829035722712t_real @ F @ I6 )
        = ( groups4148127829035722712t_real @ G @ I6 ) )
     => ( ! [I3: extended_enat] :
            ( ( member_Extended_enat @ I3 @ I6 )
           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_Extended_enat @ I @ I6 )
         => ( ( finite4001608067531595151d_enat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5184_sum__mono__inv,axiom,
    ! [F: real > nat,I6: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I6 )
        = ( groups1935376822645274424al_nat @ G @ I6 ) )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real2 @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5185_sum__mono__inv,axiom,
    ! [F: $o > nat,I6: set_o,G: $o > nat,I: $o] :
      ( ( ( groups8507830703676809646_o_nat @ F @ I6 )
        = ( groups8507830703676809646_o_nat @ G @ I6 ) )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o2 @ I @ I6 )
         => ( ( finite_finite_o @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5186_sum__mono__inv,axiom,
    ! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
        = ( groups5693394587270226106ex_nat @ G @ I6 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5187_sum__mono__inv,axiom,
    ! [F: int > nat,I6: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
        = ( groups4541462559716669496nt_nat @ G @ I6 ) )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int2 @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5188_sum__mono__inv,axiom,
    ! [F: extended_enat > nat,I6: set_Extended_enat,G: extended_enat > nat,I: extended_enat] :
      ( ( ( groups2027974829824023292at_nat @ F @ I6 )
        = ( groups2027974829824023292at_nat @ G @ I6 ) )
     => ( ! [I3: extended_enat] :
            ( ( member_Extended_enat @ I3 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_Extended_enat @ I @ I6 )
         => ( ( finite4001608067531595151d_enat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5189_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5190_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5191_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ ( suc @ X5 ) @ A2 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5192_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ ( suc @ X5 ) @ A2 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5193_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M )
        & ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_5194_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M )
        & ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_5195_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).

% ceiling_mono
thf(fact_5196_le__of__int__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_5197_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_5198_bit__of__bool__iff,axiom,
    ! [B: $o,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N2 )
      = ( B
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5199_bit__of__bool__iff,axiom,
    ! [B: $o,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N2 )
      = ( B
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_5200_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5201_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > nat,C: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2027974829824023292at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5202_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5203_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > int,C: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2025484359314973016at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2025484359314973016at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5204_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5205_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > real,C: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups4148127829035722712t_real
              @ ^ [K2: extended_enat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5206_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > extended_enat,C: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups1752964319039525884d_enat
              @ ^ [K2: complex] : ( if_Extended_enat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_p3455044024723400733d_enat @ ( B @ A ) @ ( groups1752964319039525884d_enat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups1752964319039525884d_enat
              @ ^ [K2: complex] : ( if_Extended_enat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1752964319039525884d_enat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5207_sum_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > extended_enat,C: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2433450451889696826d_enat
              @ ^ [K2: extended_enat] : ( if_Extended_enat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_p3455044024723400733d_enat @ ( B @ A ) @ ( groups2433450451889696826d_enat @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2433450451889696826d_enat
              @ ^ [K2: extended_enat] : ( if_Extended_enat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2433450451889696826d_enat @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5208_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5209_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > int,C: real > int] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_5210_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A7: set_complex] :
              ( ? [A5: complex] :
                  ( A
                  = ( insert_complex @ A5 @ A7 ) )
             => ~ ( finite3207457112153483333omplex @ A7 ) ) ) ) ).

% finite.cases
thf(fact_5211_finite_Ocases,axiom,
    ! [A: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A )
     => ( ( A != bot_bo7653980558646680370d_enat )
       => ~ ! [A7: set_Extended_enat] :
              ( ? [A5: extended_enat] :
                  ( A
                  = ( insert_Extended_enat @ A5 @ A7 ) )
             => ~ ( finite4001608067531595151d_enat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_5212_finite_Ocases,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ~ ! [A7: set_real] :
              ( ? [A5: real] :
                  ( A
                  = ( insert_real2 @ A5 @ A7 ) )
             => ~ ( finite_finite_real @ A7 ) ) ) ) ).

% finite.cases
thf(fact_5213_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A5: $o] :
                  ( A
                  = ( insert_o2 @ A5 @ A7 ) )
             => ~ ( finite_finite_o @ A7 ) ) ) ) ).

% finite.cases
thf(fact_5214_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A5: nat] :
                  ( A
                  = ( insert_nat2 @ A5 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_5215_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A5: int] :
                  ( A
                  = ( insert_int2 @ A5 @ A7 ) )
             => ~ ( finite_finite_int @ A7 ) ) ) ) ).

% finite.cases
thf(fact_5216_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A3: set_complex] :
          ( ( A3 = bot_bot_set_complex )
          | ? [A4: set_complex,B3: complex] :
              ( ( A3
                = ( insert_complex @ B3 @ A4 ) )
              & ( finite3207457112153483333omplex @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_5217_finite_Osimps,axiom,
    ( finite4001608067531595151d_enat
    = ( ^ [A3: set_Extended_enat] :
          ( ( A3 = bot_bo7653980558646680370d_enat )
          | ? [A4: set_Extended_enat,B3: extended_enat] :
              ( ( A3
                = ( insert_Extended_enat @ B3 @ A4 ) )
              & ( finite4001608067531595151d_enat @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_5218_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A3: set_real] :
          ( ( A3 = bot_bot_set_real )
          | ? [A4: set_real,B3: real] :
              ( ( A3
                = ( insert_real2 @ B3 @ A4 ) )
              & ( finite_finite_real @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_5219_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A3: set_o] :
          ( ( A3 = bot_bot_set_o )
          | ? [A4: set_o,B3: $o] :
              ( ( A3
                = ( insert_o2 @ B3 @ A4 ) )
              & ( finite_finite_o @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_5220_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A3: set_nat] :
          ( ( A3 = bot_bot_set_nat )
          | ? [A4: set_nat,B3: nat] :
              ( ( A3
                = ( insert_nat2 @ B3 @ A4 ) )
              & ( finite_finite_nat @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_5221_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A3: set_int] :
          ( ( A3 = bot_bot_set_int )
          | ? [A4: set_int,B3: int] :
              ( ( A3
                = ( insert_int2 @ B3 @ A4 ) )
              & ( finite_finite_int @ A4 ) ) ) ) ) ).

% finite.simps
thf(fact_5222_finite__induct,axiom,
    ! [F3: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_set_nat2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5223_finite__induct,axiom,
    ! [F3: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5224_finite__induct,axiom,
    ! [F3: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F4 )
             => ( ~ ( member_Extended_enat @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5225_finite__induct,axiom,
    ! [F3: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_real2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5226_finite__induct,axiom,
    ! [F3: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X5: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_o2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5227_finite__induct,axiom,
    ! [F3: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_nat2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5228_finite__induct,axiom,
    ! [F3: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_int2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5229_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X5: set_nat] : ( P2 @ ( insert_set_nat2 @ X5 @ bot_bot_set_set_nat ) )
         => ( ! [X5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat2 @ X5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_set_nat2 @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5230_finite__ne__induct,axiom,
    ! [F3: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X5: complex] : ( P2 @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
         => ( ! [X5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_complex @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5231_finite__ne__induct,axiom,
    ! [F3: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( F3 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] : ( P2 @ ( 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 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5232_finite__ne__induct,axiom,
    ! [F3: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X5: real] : ( P2 @ ( insert_real2 @ X5 @ bot_bot_set_real ) )
         => ( ! [X5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ ( member_real2 @ X5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_real2 @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5233_finite__ne__induct,axiom,
    ! [F3: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( F3 != bot_bot_set_o )
       => ( ! [X5: $o] : ( P2 @ ( insert_o2 @ X5 @ bot_bot_set_o ) )
         => ( ! [X5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ ( member_o2 @ X5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_o2 @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5234_finite__ne__induct,axiom,
    ! [F3: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X5: nat] : ( P2 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
         => ( ! [X5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ ( member_nat2 @ X5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_nat2 @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5235_finite__ne__induct,axiom,
    ! [F3: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X5: int] : ( P2 @ ( insert_int2 @ X5 @ bot_bot_set_int ) )
         => ( ! [X5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ ( member_int2 @ X5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_int2 @ X5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5236_infinite__finite__induct,axiom,
    ! [P2: set_set_nat > $o,A2: set_set_nat] :
      ( ! [A7: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_set_nat2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5237_infinite__finite__induct,axiom,
    ! [P2: set_complex > $o,A2: set_complex] :
      ( ! [A7: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5238_infinite__finite__induct,axiom,
    ! [P2: set_Extended_enat > $o,A2: set_Extended_enat] :
      ( ! [A7: set_Extended_enat] :
          ( ~ ( finite4001608067531595151d_enat @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,F4: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ F4 )
             => ( ~ ( member_Extended_enat @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5239_infinite__finite__induct,axiom,
    ! [P2: set_real > $o,A2: set_real] :
      ( ! [A7: set_real] :
          ( ~ ( finite_finite_real @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_real2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5240_infinite__finite__induct,axiom,
    ! [P2: set_o > $o,A2: set_o] :
      ( ! [A7: set_o] :
          ( ~ ( finite_finite_o @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X5: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_o2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5241_infinite__finite__induct,axiom,
    ! [P2: set_nat > $o,A2: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_nat2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5242_infinite__finite__induct,axiom,
    ! [P2: set_int > $o,A2: set_int] :
      ( ! [A7: set_int] :
          ( ~ ( finite_finite_int @ A7 )
         => ( P2 @ A7 ) )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int2 @ X5 @ F4 )
               => ( ( P2 @ F4 )
                 => ( P2 @ ( insert_int2 @ X5 @ F4 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5243_subset__singleton__iff,axiom,
    ! [X7: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X7 @ ( insert_real2 @ A @ bot_bot_set_real ) )
      = ( ( X7 = bot_bot_set_real )
        | ( X7
          = ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_5244_subset__singleton__iff,axiom,
    ! [X7: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X7 @ ( insert_o2 @ A @ bot_bot_set_o ) )
      = ( ( X7 = bot_bot_set_o )
        | ( X7
          = ( insert_o2 @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_5245_subset__singleton__iff,axiom,
    ! [X7: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X7 @ ( insert_int2 @ A @ bot_bot_set_int ) )
      = ( ( X7 = bot_bot_set_int )
        | ( X7
          = ( insert_int2 @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_5246_subset__singleton__iff,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X7 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
      = ( ( X7 = bot_bot_set_nat )
        | ( X7
          = ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_5247_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_5248_subset__singletonD,axiom,
    ! [A2: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) )
     => ( ( A2 = bot_bot_set_o )
        | ( A2
          = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_5249_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_5250_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_5251_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( A = B )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o2 @ A @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5252_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5253_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int2 @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5254_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real2 @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5255_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat2 @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X @ A2 ) @ ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5256_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ ( member_real2 @ X @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real2 @ X @ A2 ) @ ( insert_real2 @ X @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5257_Diff__insert__absorb,axiom,
    ! [X: $o,A2: set_o] :
      ( ~ ( member_o2 @ X @ A2 )
     => ( ( minus_minus_set_o @ ( insert_o2 @ X @ A2 ) @ ( insert_o2 @ X @ bot_bot_set_o ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5258_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ ( member_int2 @ X @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int2 @ X @ A2 ) @ ( insert_int2 @ X @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5259_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ ( member_nat2 @ X @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5260_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5261_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5262_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5263_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5264_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat2 @ A @ A2 )
     => ( ( insert_set_nat2 @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5265_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real2 @ A @ A2 )
     => ( ( insert_real2 @ A @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5266_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o2 @ A @ A2 )
     => ( ( insert_o2 @ A @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5267_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int2 @ A @ A2 )
     => ( ( insert_int2 @ A @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5268_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat2 @ A @ A2 )
     => ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5269_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_5270_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o2 @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_5271_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int2 @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_5272_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_5273_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X: real,C4: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real2 @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C4 ) )
        & ~ ( member_real2 @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5274_subset__Diff__insert,axiom,
    ! [A2: set_o,B2: set_o,X: $o,C4: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ ( insert_o2 @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ C4 ) )
        & ~ ( member_o2 @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5275_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat2 @ X @ C4 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C4 ) )
        & ~ ( member_set_nat2 @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5276_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int2 @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C4 ) )
        & ~ ( member_int2 @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5277_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat2 @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C4 ) )
        & ~ ( member_nat2 @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5278_sum_Ointer__filter,axiom,
    ! [A2: set_o,G: $o > nat,P2: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8507830703676809646_o_nat @ G
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups8507830703676809646_o_nat
          @ ^ [X2: $o] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5279_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > nat,P2: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X2: real] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5280_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > nat,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups5693394587270226106ex_nat
          @ ^ [X2: complex] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5281_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > nat,P2: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X2: int] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5282_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,P2: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X2: extended_enat] :
                ( ( member_Extended_enat @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups2027974829824023292at_nat
          @ ^ [X2: extended_enat] : ( if_nat @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5283_sum_Ointer__filter,axiom,
    ! [A2: set_o,G: $o > real,P2: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8691415230153176458o_real @ G
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups8691415230153176458o_real
          @ ^ [X2: $o] : ( if_real @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5284_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P2: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X2: real] : ( if_real @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5285_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X2: complex] : ( if_real @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5286_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P2: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X2: int] : ( if_real @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5287_sum_Ointer__filter,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real,P2: extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G
          @ ( collec4429806609662206161d_enat
            @ ^ [X2: extended_enat] :
                ( ( member_Extended_enat @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( groups4148127829035722712t_real
          @ ^ [X2: extended_enat] : ( if_real @ ( P2 @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_5288_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > extended_enat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups1752964319039525884d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5289_member__le__sum,axiom,
    ! [I: extended_enat,A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( member_Extended_enat @ I @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite4001608067531595151d_enat @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups2433450451889696826d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5290_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > extended_enat] :
      ( ( member_real2 @ I @ A2 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups2800946370649118462d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5291_member__le__sum,axiom,
    ! [I: $o,A2: set_o,F: $o > extended_enat] :
      ( ( member_o2 @ I @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups7198740251461348360d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5292_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > extended_enat] :
      ( ( member_int2 @ I @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups4225252721152677374d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5293_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > extended_enat] :
      ( ( member_nat2 @ I @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ I @ bot_bot_set_nat ) ) )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ ( groups7108830773950497114d_enat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5294_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5295_member__le__sum,axiom,
    ! [I: extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( member_Extended_enat @ I @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite4001608067531595151d_enat @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups4148127829035722712t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5296_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real2 @ I @ A2 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5297_member__le__sum,axiom,
    ! [I: $o,A2: set_o,F: $o > real] :
      ( ( member_o2 @ I @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8691415230153176458o_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_5298_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5299_sum__subtractf__nat,axiom,
    ! [A2: set_o,G: $o > nat,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups8507830703676809646_o_nat
          @ ^ [X2: $o] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5300_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X2: set_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5301_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5302_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5303_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5304_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5305_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5306_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5307_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > extended_enat,I: nat > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat2 @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat2 @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat2 @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups7108830773950497114d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5308_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > extended_enat,I: complex > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat2 @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups1752964319039525884d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5309_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > extended_enat,I: int > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat2 @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int2 @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups4225252721152677374d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5310_sum__le__included,axiom,
    ! [S: set_nat,T: set_Extended_enat,G: extended_enat > extended_enat,I: extended_enat > nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat2 @ X5 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups7108830773950497114d_enat @ F @ S ) @ ( groups2433450451889696826d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5311_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: nat > extended_enat,I: nat > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat2 @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat2 @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups7108830773950497114d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5312_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > extended_enat,I: complex > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups1752964319039525884d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5313_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > extended_enat,I: int > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int2 @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups4225252721152677374d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5314_sum__le__included,axiom,
    ! [S: set_complex,T: set_Extended_enat,G: extended_enat > extended_enat,I: extended_enat > complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite4001608067531595151d_enat @ T )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: extended_enat] :
                    ( ( member_Extended_enat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ S ) @ ( groups2433450451889696826d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5315_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > extended_enat,I: nat > int,F: int > extended_enat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat2 @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat2 @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ S ) @ ( groups7108830773950497114d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5316_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > extended_enat,I: complex > int,F: int > extended_enat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ S ) @ ( groups1752964319039525884d_enat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5317_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups2800946370649118462d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X2: real] :
                ( ( member_real2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5318_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: $o > extended_enat] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups7198740251461348360d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X2: $o] :
                ( ( member_o2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5319_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups7108830773950497114d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X2: nat] :
                ( ( member_nat2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5320_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups1752964319039525884d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5321_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups4225252721152677374d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X2: int] :
                ( ( member_int2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5322_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) ) )
       => ( ( ( groups2433450451889696826d_enat @ F @ A2 )
            = zero_z5237406670263579293d_enat )
          = ( ! [X2: extended_enat] :
                ( ( member_Extended_enat @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z5237406670263579293d_enat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5323_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: real] :
                ( ( member_real2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5324_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: $o] :
                ( ( member_o2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5325_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5326_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: int] :
                ( ( member_int2 @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5327_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5328_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int2 @ X3 @ A2 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5329_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5330_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5331_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int2 @ X3 @ A2 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5332_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5333_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat2 @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5334_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5335_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int2 @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5336_sum__strict__mono__ex1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: extended_enat] :
              ( ( member_Extended_enat @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( groups2025484359314973016at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5337_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5338_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5339_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2027974829824023292at_nat @ H2 @ S3 ) @ ( groups2027974829824023292at_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5340_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5341_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5342_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4148127829035722712t_real @ H2 @ S3 ) @ ( groups4148127829035722712t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5343_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat2 @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5344_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_complex,H2: complex > int,G: complex > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5690904116761175830ex_int @ H2 @ S3 ) @ ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5345_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_int,H2: int > int,G: int > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4538972089207619220nt_int @ H2 @ S3 ) @ ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5346_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_Extended_enat,H2: extended_enat > int,G: extended_enat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X15 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite4001608067531595151d_enat @ S3 )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2025484359314973016at_int @ H2 @ S3 ) @ ( groups2025484359314973016at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5347_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5348_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5349_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5350_sum__strict__mono,axiom,
    ! [A2: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5351_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5352_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > extended_enat,G: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups1752964319039525884d_enat @ F @ A2 ) @ ( groups1752964319039525884d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5353_sum__strict__mono,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > extended_enat,G: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups2433450451889696826d_enat @ F @ A2 ) @ ( groups2433450451889696826d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5354_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > extended_enat,G: real > extended_enat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) @ ( groups2800946370649118462d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5355_sum__strict__mono,axiom,
    ! [A2: set_o,F: $o > extended_enat,G: $o > extended_enat] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups7198740251461348360d_enat @ F @ A2 ) @ ( groups7198740251461348360d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5356_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > extended_enat,G: nat > extended_enat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( ( member_nat2 @ X5 @ A2 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_le72135733267957522d_enat @ ( groups7108830773950497114d_enat @ F @ A2 ) @ ( groups7108830773950497114d_enat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_5357_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_real2 @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real2 @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_real2 @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real2 @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: real] :
                          ( ( member_real2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S3 )
                        = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5358_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_o,S3: set_real,I: $o > real,J: real > $o,T3: set_o,G: real > nat,H2: $o > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_o @ T4 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_o2 @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T4 ) ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: $o] :
                    ( ( member_o2 @ B4 @ ( minus_minus_set_o @ T3 @ T4 ) )
                   => ( member_real2 @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: $o] :
                        ( ( member_o2 @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: real] :
                          ( ( member_real2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S3 )
                        = ( groups8507830703676809646_o_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5359_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T4: set_real,S3: set_o,I: real > $o,J: $o > real,T3: set_real,G: $o > nat,H2: real > nat] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
               => ( member_real2 @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: real] :
                    ( ( member_real2 @ B4 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_o2 @ ( I @ B4 ) @ ( minus_minus_set_o @ S3 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: real] :
                        ( ( member_real2 @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: $o] :
                          ( ( member_o2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8507830703676809646_o_nat @ G @ S3 )
                        = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5360_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T4: set_o,S3: set_o,I: $o > $o,J: $o > $o,T3: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_o @ T4 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
               => ( member_o2 @ ( J @ A5 ) @ ( minus_minus_set_o @ T3 @ T4 ) ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: $o] :
                    ( ( member_o2 @ B4 @ ( minus_minus_set_o @ T3 @ T4 ) )
                   => ( member_o2 @ ( I @ B4 ) @ ( minus_minus_set_o @ S3 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: $o] :
                        ( ( member_o2 @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: $o] :
                          ( ( member_o2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8507830703676809646_o_nat @ G @ S3 )
                        = ( groups8507830703676809646_o_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5361_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > nat,H2: complex > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_real2 @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: real] :
                          ( ( member_real2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S3 )
                        = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5362_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T4: set_complex,S3: set_o,I: complex > $o,J: $o > complex,T3: set_complex,G: $o > nat,H2: complex > nat] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: complex] :
                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_o2 @ ( I @ B4 ) @ ( minus_minus_set_o @ S3 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: $o] :
                          ( ( member_o2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8507830703676809646_o_nat @ G @ S3 )
                        = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5363_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > nat,H2: int > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_int2 @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int2 @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_real2 @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: int] :
                        ( ( member_int2 @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: real] :
                          ( ( member_real2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S3 )
                        = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5364_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T4: set_int,S3: set_o,I: int > $o,J: $o > int,T3: set_int,G: $o > nat,H2: int > nat] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
               => ( member_int2 @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: int] :
                    ( ( member_int2 @ B4 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_o2 @ ( I @ B4 ) @ ( minus_minus_set_o @ S3 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: int] :
                        ( ( member_int2 @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: $o] :
                          ( ( member_o2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8507830703676809646_o_nat @ G @ S3 )
                        = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5365_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_Extended_enat,S3: set_real,I: extended_enat > real,J: real > extended_enat,T3: set_Extended_enat,G: real > nat,H2: extended_enat > nat] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite4001608067531595151d_enat @ T4 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A5 ) @ ( minus_925952699566721837d_enat @ T3 @ T4 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T4 ) )
                   => ( member_real2 @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: real] :
                          ( ( member_real2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups1935376822645274424al_nat @ G @ S3 )
                        = ( groups2027974829824023292at_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5366_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_o,T4: set_Extended_enat,S3: set_o,I: extended_enat > $o,J: $o > extended_enat,T3: set_Extended_enat,G: $o > nat,H2: extended_enat > nat] :
      ( ( finite_finite_o @ S5 )
     => ( ( finite4001608067531595151d_enat @ T4 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ S3 @ S5 ) )
               => ( member_Extended_enat @ ( J @ A5 ) @ ( minus_925952699566721837d_enat @ T3 @ T4 ) ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B4 ) )
                    = B4 ) )
             => ( ! [B4: extended_enat] :
                    ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ T3 @ T4 ) )
                   => ( member_o2 @ ( I @ B4 ) @ ( minus_minus_set_o @ S3 @ S5 ) ) )
               => ( ! [A5: $o] :
                      ( ( member_o2 @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_nat ) )
                 => ( ! [B4: extended_enat] :
                        ( ( member_Extended_enat @ B4 @ T4 )
                       => ( ( H2 @ B4 )
                          = zero_zero_nat ) )
                   => ( ! [A5: $o] :
                          ( ( member_o2 @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8507830703676809646_o_nat @ G @ S3 )
                        = ( groups2027974829824023292at_nat @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5367_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5368_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: int] :
              ( ( member_int2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: int] :
                  ( ( member_int2 @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5369_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: extended_enat] :
                  ( ( member_Extended_enat @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5370_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: nat] :
              ( ( member_nat2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: nat] :
                  ( ( member_nat2 @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5371_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N2 ) )
     => ? [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_5372_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5373_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: int] :
              ( ( member_int2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: int] :
                  ( ( member_int2 @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5374_sum__eq__1__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2027974829824023292at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: extended_enat] :
                  ( ( member_Extended_enat @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5375_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: nat] :
              ( ( member_nat2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: nat] :
                  ( ( member_nat2 @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5376_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).

% ceiling_le
thf(fact_5377_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5378_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5379_ceiling__add__le,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_5380_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X5: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_complex @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5381_finite__ranking__induct,axiom,
    ! [S3: set_Extended_enat,P2: set_Extended_enat > $o,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,S6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S6 )
             => ( ! [Y4: extended_enat] :
                    ( ( member_Extended_enat @ Y4 @ S6 )
                   => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_Extended_enat @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5382_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: set_real > $o,F: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X5: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y4: real] :
                    ( ( member_real2 @ Y4 @ S6 )
                   => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_real2 @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5383_finite__ranking__induct,axiom,
    ! [S3: set_o,P2: set_o > $o,F: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X5: $o,S6: set_o] :
              ( ( finite_finite_o @ S6 )
             => ( ! [Y4: $o] :
                    ( ( member_o2 @ Y4 @ S6 )
                   => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_o2 @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5384_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: set_nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X5: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y4: nat] :
                    ( ( member_nat2 @ Y4 @ S6 )
                   => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_nat2 @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5385_finite__ranking__induct,axiom,
    ! [S3: set_int,P2: set_int > $o,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X5: int,S6: set_int] :
              ( ( finite_finite_int @ S6 )
             => ( ! [Y4: int] :
                    ( ( member_int2 @ Y4 @ S6 )
                   => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_int2 @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5386_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X5: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_complex @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5387_finite__ranking__induct,axiom,
    ! [S3: set_Extended_enat,P2: set_Extended_enat > $o,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [X5: extended_enat,S6: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ S6 )
             => ( ! [Y4: extended_enat] :
                    ( ( member_Extended_enat @ Y4 @ S6 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_Extended_enat @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5388_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: set_real > $o,F: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X5: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y4: real] :
                    ( ( member_real2 @ Y4 @ S6 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_real2 @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5389_finite__ranking__induct,axiom,
    ! [S3: set_o,P2: set_o > $o,F: $o > nat] :
      ( ( finite_finite_o @ S3 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [X5: $o,S6: set_o] :
              ( ( finite_finite_o @ S6 )
             => ( ! [Y4: $o] :
                    ( ( member_o2 @ Y4 @ S6 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X5 ) ) )
               => ( ( P2 @ S6 )
                 => ( P2 @ ( insert_o2 @ X5 @ S6 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5390_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [B4: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X3: $o] :
                    ( ( member_o2 @ X3 @ A7 )
                   => ( ord_less_o @ X3 @ B4 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_o2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5391_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [B4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X3: nat] :
                    ( ( member_nat2 @ X3 @ A7 )
                   => ( ord_less_nat @ X3 @ B4 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_nat2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5392_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [B4: extended_enat,A7: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A7 )
             => ( ! [X3: extended_enat] :
                    ( ( member_Extended_enat @ X3 @ A7 )
                   => ( ord_le72135733267957522d_enat @ X3 @ B4 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_Extended_enat @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5393_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [B4: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X3: real] :
                    ( ( member_real2 @ X3 @ A7 )
                   => ( ord_less_real @ X3 @ B4 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_real2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5394_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [B4: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X3: int] :
                    ( ( member_int2 @ X3 @ A7 )
                   => ( ord_less_int @ X3 @ B4 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_int2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5395_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [B4: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X3: $o] :
                    ( ( member_o2 @ X3 @ A7 )
                   => ( ord_less_o @ B4 @ X3 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_o2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5396_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [B4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X3: nat] :
                    ( ( member_nat2 @ X3 @ A7 )
                   => ( ord_less_nat @ B4 @ X3 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_nat2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5397_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [B4: extended_enat,A7: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A7 )
             => ( ! [X3: extended_enat] :
                    ( ( member_Extended_enat @ X3 @ A7 )
                   => ( ord_le72135733267957522d_enat @ B4 @ X3 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_Extended_enat @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5398_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [B4: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X3: real] :
                    ( ( member_real2 @ X3 @ A7 )
                   => ( ord_less_real @ B4 @ X3 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_real2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5399_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [B4: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X3: int] :
                    ( ( member_int2 @ X3 @ A7 )
                   => ( ord_less_int @ B4 @ X3 ) )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( insert_int2 @ B4 @ A7 ) ) ) ) )
         => ( P2 @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5400_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat2 @ A5 @ A2 )
                 => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
                   => ( ~ ( member_set_nat2 @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_set_nat2 @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5401_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_complex )
         => ( ! [A5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
                   => ( ~ ( member_complex @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_complex @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5402_finite__subset__induct_H,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ord_le7203529160286727270d_enat @ F3 @ A2 )
       => ( ( P2 @ bot_bo7653980558646680370d_enat )
         => ( ! [A5: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( member_Extended_enat @ A5 @ A2 )
                 => ( ( ord_le7203529160286727270d_enat @ F4 @ A2 )
                   => ( ~ ( member_Extended_enat @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_Extended_enat @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5403_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_real )
         => ( ! [A5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real2 @ A5 @ A2 )
                 => ( ( ord_less_eq_set_real @ F4 @ A2 )
                   => ( ~ ( member_real2 @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_real2 @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5404_finite__subset__induct_H,axiom,
    ! [F3: set_o,A2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( ord_less_eq_set_o @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_o )
         => ( ! [A5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o2 @ A5 @ A2 )
                 => ( ( ord_less_eq_set_o @ F4 @ A2 )
                   => ( ~ ( member_o2 @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_o2 @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5405_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_int )
         => ( ! [A5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int2 @ A5 @ A2 )
                 => ( ( ord_less_eq_set_int @ F4 @ A2 )
                   => ( ~ ( member_int2 @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_int2 @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5406_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat2 @ A5 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F4 @ A2 )
                   => ( ~ ( member_nat2 @ A5 @ F4 )
                     => ( ( P2 @ F4 )
                       => ( P2 @ ( insert_nat2 @ A5 @ F4 ) ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5407_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat2 @ A5 @ A2 )
                 => ( ~ ( member_set_nat2 @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_set_nat2 @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5408_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_complex )
         => ( ! [A5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ~ ( member_complex @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_complex @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5409_finite__subset__induct,axiom,
    ! [F3: set_Extended_enat,A2: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ord_le7203529160286727270d_enat @ F3 @ A2 )
       => ( ( P2 @ bot_bo7653980558646680370d_enat )
         => ( ! [A5: extended_enat,F4: set_Extended_enat] :
                ( ( finite4001608067531595151d_enat @ F4 )
               => ( ( member_Extended_enat @ A5 @ A2 )
                 => ( ~ ( member_Extended_enat @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_Extended_enat @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5410_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_real )
         => ( ! [A5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real2 @ A5 @ A2 )
                 => ( ~ ( member_real2 @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_real2 @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5411_finite__subset__induct,axiom,
    ! [F3: set_o,A2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( ord_less_eq_set_o @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_o )
         => ( ! [A5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o2 @ A5 @ A2 )
                 => ( ~ ( member_o2 @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_o2 @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5412_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_int )
         => ( ! [A5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int2 @ A5 @ A2 )
                 => ( ~ ( member_int2 @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_int2 @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5413_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat2 @ A5 @ A2 )
                 => ( ~ ( member_nat2 @ A5 @ F4 )
                   => ( ( P2 @ F4 )
                     => ( P2 @ ( insert_nat2 @ A5 @ F4 ) ) ) ) ) )
           => ( P2 @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5414_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > extended_enat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups2800946370649118462d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_real2 @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5415_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > extended_enat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups7198740251461348360d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_o2 @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5416_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > extended_enat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups7108830773950497114d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_nat2 @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5417_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > extended_enat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups1752964319039525884d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5418_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > extended_enat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups4225252721152677374d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_int2 @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5419_sum__nonneg__0,axiom,
    ! [S: set_Extended_enat,F: extended_enat > extended_enat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I3: extended_enat] :
            ( ( member_Extended_enat @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups2433450451889696826d_enat @ F @ S )
            = zero_z5237406670263579293d_enat )
         => ( ( member_Extended_enat @ I @ S )
           => ( ( F @ I )
              = zero_z5237406670263579293d_enat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5420_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real2 @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5421_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > real,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S )
            = zero_zero_real )
         => ( ( member_o2 @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5422_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5423_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int2 @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5424_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > extended_enat,B2: extended_enat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups2800946370649118462d_enat @ F @ S )
            = B2 )
         => ( ( member_real2 @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5425_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > extended_enat,B2: extended_enat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups7198740251461348360d_enat @ F @ S )
            = B2 )
         => ( ( member_o2 @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5426_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > extended_enat,B2: extended_enat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups7108830773950497114d_enat @ F @ S )
            = B2 )
         => ( ( member_nat2 @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5427_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > extended_enat,B2: extended_enat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups1752964319039525884d_enat @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5428_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > extended_enat,B2: extended_enat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups4225252721152677374d_enat @ F @ S )
            = B2 )
         => ( ( member_int2 @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5429_sum__nonneg__leq__bound,axiom,
    ! [S: set_Extended_enat,F: extended_enat > extended_enat,B2: extended_enat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ S )
     => ( ! [I3: extended_enat] :
            ( ( member_Extended_enat @ I3 @ S )
           => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
       => ( ( ( groups2433450451889696826d_enat @ F @ S )
            = B2 )
         => ( ( member_Extended_enat @ I @ S )
           => ( ord_le2932123472753598470d_enat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5430_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B2: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B2 )
         => ( ( member_real2 @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5431_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > real,B2: real,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S )
            = B2 )
         => ( ( member_o2 @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5432_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B2: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5433_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B2: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B2 )
         => ( ( member_int2 @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5434_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_5435_infinite__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ S3 )
     => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% infinite_remove
thf(fact_5436_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ ( finite_finite_real @ S3 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_5437_infinite__remove,axiom,
    ! [S3: set_o,A: $o] :
      ( ~ ( finite_finite_o @ S3 )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S3 @ ( insert_o2 @ A @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_5438_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ ( finite_finite_int @ S3 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int2 @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_5439_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_5440_infinite__coinduct,axiom,
    ! [X7: set_complex > $o,A2: set_complex] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_complex] :
            ( ( X7 @ A7 )
           => ? [X3: complex] :
                ( ( member_complex @ X3 @ A7 )
                & ( ( X7 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5441_infinite__coinduct,axiom,
    ! [X7: set_Extended_enat > $o,A2: set_Extended_enat] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_Extended_enat] :
            ( ( X7 @ A7 )
           => ? [X3: extended_enat] :
                ( ( member_Extended_enat @ X3 @ A7 )
                & ( ( X7 @ ( minus_925952699566721837d_enat @ A7 @ ( insert_Extended_enat @ X3 @ bot_bo7653980558646680370d_enat ) ) )
                  | ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A7 @ ( insert_Extended_enat @ X3 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
       => ~ ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5442_infinite__coinduct,axiom,
    ! [X7: set_real > $o,A2: set_real] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_real] :
            ( ( X7 @ A7 )
           => ? [X3: real] :
                ( ( member_real2 @ X3 @ A7 )
                & ( ( X7 @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X3 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X3 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5443_infinite__coinduct,axiom,
    ! [X7: set_o > $o,A2: set_o] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_o] :
            ( ( X7 @ A7 )
           => ? [X3: $o] :
                ( ( member_o2 @ X3 @ A7 )
                & ( ( X7 @ ( minus_minus_set_o @ A7 @ ( insert_o2 @ X3 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A7 @ ( insert_o2 @ X3 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5444_infinite__coinduct,axiom,
    ! [X7: set_int > $o,A2: set_int] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_int] :
            ( ( X7 @ A7 )
           => ? [X3: int] :
                ( ( member_int2 @ X3 @ A7 )
                & ( ( X7 @ ( minus_minus_set_int @ A7 @ ( insert_int2 @ X3 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int2 @ X3 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5445_infinite__coinduct,axiom,
    ! [X7: set_nat > $o,A2: set_nat] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_nat] :
            ( ( X7 @ A7 )
           => ? [X3: nat] :
                ( ( member_nat2 @ X3 @ A7 )
                & ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5446_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: set_nat,A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( member_set_nat2 @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat2 @ A5 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P2 @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5447_finite__empty__induct,axiom,
    ! [A2: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: complex,A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( member_complex @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A5 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P2 @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_5448_finite__empty__induct,axiom,
    ! [A2: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: extended_enat,A7: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A7 )
             => ( ( member_Extended_enat @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_925952699566721837d_enat @ A7 @ ( insert_Extended_enat @ A5 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
         => ( P2 @ bot_bo7653980558646680370d_enat ) ) ) ) ).

% finite_empty_induct
thf(fact_5449_finite__empty__induct,axiom,
    ! [A2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( member_real2 @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ A5 @ bot_bot_set_real ) ) ) ) ) )
         => ( P2 @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_5450_finite__empty__induct,axiom,
    ! [A2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( member_o2 @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_minus_set_o @ A7 @ ( insert_o2 @ A5 @ bot_bot_set_o ) ) ) ) ) )
         => ( P2 @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_5451_finite__empty__induct,axiom,
    ! [A2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( member_int2 @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_minus_set_int @ A7 @ ( insert_int2 @ A5 @ bot_bot_set_int ) ) ) ) ) )
         => ( P2 @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_5452_finite__empty__induct,axiom,
    ! [A2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P2 @ A2 )
       => ( ! [A5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat2 @ A5 @ A7 )
               => ( ( P2 @ A7 )
                 => ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P2 @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5453_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5454_Diff__single__insert,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ B2 )
     => ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5455_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5456_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5457_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat2 @ X @ B2 ) )
      = ( ( ( member_set_nat2 @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
        & ( ~ ( member_set_nat2 @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5458_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real2 @ X @ B2 ) )
      = ( ( ( member_real2 @ X @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) @ B2 ) )
        & ( ~ ( member_real2 @ X @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5459_subset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ X @ B2 ) )
      = ( ( ( member_o2 @ X @ A2 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ B2 ) )
        & ( ~ ( member_o2 @ X @ A2 )
         => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5460_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int2 @ X @ B2 ) )
      = ( ( ( member_int2 @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ B2 ) )
        & ( ~ ( member_int2 @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5461_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
      = ( ( ( member_nat2 @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
        & ( ~ ( member_nat2 @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5462_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat2 @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_5463_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( insert_nat2 @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_5464_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
        = ( insert_nat2 @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_5465_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M @ N2 )
        = ( insert_nat2 @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_5466_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5467_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5468_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5469_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2027974829824023292at_nat @ G
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X2: extended_enat] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups2027974829824023292at_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5470_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5471_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5472_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5473_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups4148127829035722712t_real @ G
          @ ( minus_925952699566721837d_enat @ A2
            @ ( collec4429806609662206161d_enat
              @ ^ [X2: extended_enat] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups4148127829035722712t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5474_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( G @ X2 )
                  = zero_zero_int ) ) ) )
        = ( groups1932886352136224148al_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5475_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5476_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real2 @ X @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5477_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X ) ) @ ( insert_o2 @ X @ ( set_o2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5478_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5479_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5480_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5481_Compl__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real2 @ X @ A2 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_5482_Compl__insert,axiom,
    ! [X: $o,A2: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o2 @ X @ A2 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_5483_Compl__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int2 @ X @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int2 @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_5484_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat2 @ X @ A2 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_5485_sum__power__add,axiom,
    ! [X: int,M: nat,I6: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5486_sum__power__add,axiom,
    ! [X: complex,M: nat,I6: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I5: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5487_sum__power__add,axiom,
    ! [X: real,M: nat,I6: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5488_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5489_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5490_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% less_mask
thf(fact_5491_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > extended_enat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real2 @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real2 @ I3 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5492_sum__pos2,axiom,
    ! [I6: set_o,I: $o,F: $o > extended_enat] :
      ( ( finite_finite_o @ I6 )
     => ( ( member_o2 @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o2 @ I3 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7198740251461348360d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5493_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat2 @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat2 @ I3 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5494_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups1752964319039525884d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5495_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > extended_enat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int2 @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int2 @ I3 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups4225252721152677374d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5496_sum__pos2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I ) )
         => ( ! [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2433450451889696826d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5497_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real2 @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real2 @ I3 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5498_sum__pos2,axiom,
    ! [I6: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I6 )
     => ( ( member_o2 @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o2 @ I3 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5499_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5500_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int2 @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int2 @ I3 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5501_sum__pos,axiom,
    ! [I6: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5502_sum__pos,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups2027974829824023292at_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5503_sum__pos,axiom,
    ! [I6: set_real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real2 @ I3 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5504_sum__pos,axiom,
    ! [I6: set_o,F: $o > nat] :
      ( ( finite_finite_o @ I6 )
     => ( ( I6 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o2 @ I3 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5505_sum__pos,axiom,
    ! [I6: set_int,F: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int2 @ I3 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5506_sum__pos,axiom,
    ! [I6: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups1752964319039525884d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5507_sum__pos,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2433450451889696826d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5508_sum__pos,axiom,
    ! [I6: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real2 @ I3 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5509_sum__pos,axiom,
    ! [I6: set_o,F: $o > extended_enat] :
      ( ( finite_finite_o @ I6 )
     => ( ( I6 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o2 @ I3 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7198740251461348360d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5510_sum__pos,axiom,
    ! [I6: set_nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat2 @ I3 @ I6 )
             => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) ) )
         => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_5511_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5512_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ T3 )
              = ( groups8507830703676809646_o_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5513_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
              = ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5514_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ T3 )
              = ( groups4541462559716669496nt_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5515_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2027974829824023292at_nat @ G @ T3 )
              = ( groups2027974829824023292at_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5516_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5517_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8691415230153176458o_real @ G @ T3 )
              = ( groups8691415230153176458o_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5518_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5519_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T3 )
              = ( groups8778361861064173332t_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5520_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4148127829035722712t_real @ G @ T3 )
              = ( groups4148127829035722712t_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5521_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S3 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5522_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H2: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ S3 )
              = ( groups8507830703676809646_o_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5523_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ S3 )
              = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5524_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ S3 )
              = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5525_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2027974829824023292at_nat @ G @ S3 )
              = ( groups2027974829824023292at_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5526_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5527_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H2: $o > real,G: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8691415230153176458o_real @ G @ S3 )
              = ( groups8691415230153176458o_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5528_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5529_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S3 )
              = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5530_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4148127829035722712t_real @ G @ S3 )
              = ( groups4148127829035722712t_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5531_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5532_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T3 )
            = ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5533_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G @ T3 )
            = ( groups2027974829824023292at_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5534_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5535_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T3 )
            = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5536_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G @ T3 )
            = ( groups4148127829035722712t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5537_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5538_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ T3 )
            = ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5539_sum_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups2025484359314973016at_int @ G @ T3 )
            = ( groups2025484359314973016at_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5540_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ T3 )
            = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5541_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5542_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S3 )
            = ( groups4541462559716669496nt_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5543_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups2027974829824023292at_nat @ G @ S3 )
            = ( groups2027974829824023292at_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5544_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5545_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S3 )
            = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5546_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups4148127829035722712t_real @ G @ S3 )
            = ( groups4148127829035722712t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5547_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5548_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ S3 )
            = ( groups4538972089207619220nt_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5549_sum_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups2025484359314973016at_int @ G @ S3 )
            = ( groups2025484359314973016at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5550_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ S3 )
            = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5551_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5552_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ C4 )
                  = ( groups8507830703676809646_o_nat @ H2 @ C4 ) )
               => ( ( groups8507830703676809646_o_nat @ G @ A2 )
                  = ( groups8507830703676809646_o_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5553_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
               => ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5554_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) )
               => ( ( groups4541462559716669496nt_nat @ G @ A2 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5555_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups2027974829824023292at_nat @ G @ C4 )
                  = ( groups2027974829824023292at_nat @ H2 @ C4 ) )
               => ( ( groups2027974829824023292at_nat @ G @ A2 )
                  = ( groups2027974829824023292at_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5556_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5557_sum_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8691415230153176458o_real @ G @ C4 )
                  = ( groups8691415230153176458o_real @ H2 @ C4 ) )
               => ( ( groups8691415230153176458o_real @ G @ A2 )
                  = ( groups8691415230153176458o_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5558_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5559_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) )
               => ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5560_sum_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G @ C4 )
                  = ( groups4148127829035722712t_real @ H2 @ C4 ) )
               => ( ( groups4148127829035722712t_real @ G @ A2 )
                  = ( groups4148127829035722712t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5561_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5562_sum_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ A2 )
                  = ( groups8507830703676809646_o_nat @ H2 @ B2 ) )
                = ( ( groups8507830703676809646_o_nat @ G @ C4 )
                  = ( groups8507830703676809646_o_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5563_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B2 ) )
                = ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5564_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B2 ) )
                = ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5565_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups2027974829824023292at_nat @ G @ A2 )
                  = ( groups2027974829824023292at_nat @ H2 @ B2 ) )
                = ( ( groups2027974829824023292at_nat @ G @ C4 )
                  = ( groups2027974829824023292at_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5566_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5567_sum_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > real,H2: $o > real] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8691415230153176458o_real @ G @ A2 )
                  = ( groups8691415230153176458o_real @ H2 @ B2 ) )
                = ( ( groups8691415230153176458o_real @ G @ C4 )
                  = ( groups8691415230153176458o_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5568_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5569_sum_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B2 ) )
                = ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5570_sum_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > real,H2: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups4148127829035722712t_real @ G @ A2 )
                  = ( groups4148127829035722712t_real @ H2 @ B2 ) )
                = ( ( groups4148127829035722712t_real @ G @ C4 )
                  = ( groups4148127829035722712t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5571_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5572_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5573_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2027974829824023292at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5574_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5575_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ A2 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5576_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > int] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2025484359314973016at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups2025484359314973016at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2025484359314973016at_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5577_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5578_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5579_sum_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups4148127829035722712t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups4148127829035722712t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups4148127829035722712t_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5580_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > extended_enat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups1752964319039525884d_enat @ G @ A2 )
          = ( plus_p3455044024723400733d_enat @ ( groups1752964319039525884d_enat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups1752964319039525884d_enat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5581_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5582_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5583_sum__diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
       => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A2 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5584_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5585_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5586_sum__diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
       => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5587_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5588_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5589_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_5590_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_5591_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_5592_sum__diff__nat,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5593_sum__diff__nat,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5594_sum__diff__nat,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
       => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5595_sum__diff__nat,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5596_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_5597_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5598_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > extended_enat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z5237406670263579293d_enat )
     => ( ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5599_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_5600_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_5601_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5602_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5603_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5604_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5605_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5606_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ ( suc @ N2 ) ) @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5607_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5608_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5609_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5610_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ M ) @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5611_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5612_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5613_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_5614_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_5615_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
                 => ( ! [X3: set_nat] :
                        ( ( member_set_nat2 @ X3 @ A7 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat2 @ X3 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5616_finite__remove__induct,axiom,
    ! [B2: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
                 => ( ! [X3: complex] :
                        ( ( member_complex @ X3 @ A7 )
                       => ( P2 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5617_finite__remove__induct,axiom,
    ! [B2: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [A7: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A7 )
             => ( ( A7 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A7 @ B2 )
                 => ( ! [X3: extended_enat] :
                        ( ( member_Extended_enat @ X3 @ A7 )
                       => ( P2 @ ( minus_925952699566721837d_enat @ A7 @ ( insert_Extended_enat @ X3 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5618_finite__remove__induct,axiom,
    ! [B2: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B2 )
                 => ( ! [X3: real] :
                        ( ( member_real2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X3 @ bot_bot_set_real ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5619_finite__remove__induct,axiom,
    ! [B2: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B2 )
                 => ( ! [X3: $o] :
                        ( ( member_o2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_o @ A7 @ ( insert_o2 @ X3 @ bot_bot_set_o ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5620_finite__remove__induct,axiom,
    ! [B2: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B2 )
                 => ( ! [X3: int] :
                        ( ( member_int2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_int @ A7 @ ( insert_int2 @ X3 @ bot_bot_set_int ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5621_finite__remove__induct,axiom,
    ! [B2: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B2 )
                 => ( ! [X3: nat] :
                        ( ( member_nat2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5622_remove__induct,axiom,
    ! [P2: set_set_nat > $o,B2: set_set_nat] :
      ( ( P2 @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
                 => ( ! [X3: set_nat] :
                        ( ( member_set_nat2 @ X3 @ A7 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat2 @ X3 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5623_remove__induct,axiom,
    ! [P2: set_complex > $o,B2: set_complex] :
      ( ( P2 @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
                 => ( ! [X3: complex] :
                        ( ( member_complex @ X3 @ A7 )
                       => ( P2 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5624_remove__induct,axiom,
    ! [P2: set_Extended_enat > $o,B2: set_Extended_enat] :
      ( ( P2 @ bot_bo7653980558646680370d_enat )
     => ( ( ~ ( finite4001608067531595151d_enat @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A7 )
             => ( ( A7 != bot_bo7653980558646680370d_enat )
               => ( ( ord_le7203529160286727270d_enat @ A7 @ B2 )
                 => ( ! [X3: extended_enat] :
                        ( ( member_Extended_enat @ X3 @ A7 )
                       => ( P2 @ ( minus_925952699566721837d_enat @ A7 @ ( insert_Extended_enat @ X3 @ bot_bo7653980558646680370d_enat ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5625_remove__induct,axiom,
    ! [P2: set_real > $o,B2: set_real] :
      ( ( P2 @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B2 )
                 => ( ! [X3: real] :
                        ( ( member_real2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_real @ A7 @ ( insert_real2 @ X3 @ bot_bot_set_real ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5626_remove__induct,axiom,
    ! [P2: set_o > $o,B2: set_o] :
      ( ( P2 @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B2 )
                 => ( ! [X3: $o] :
                        ( ( member_o2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_o @ A7 @ ( insert_o2 @ X3 @ bot_bot_set_o ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5627_remove__induct,axiom,
    ! [P2: set_int > $o,B2: set_int] :
      ( ( P2 @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B2 )
                 => ( ! [X3: int] :
                        ( ( member_int2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_int @ A7 @ ( insert_int2 @ X3 @ bot_bot_set_int ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5628_remove__induct,axiom,
    ! [P2: set_nat > $o,B2: set_nat] :
      ( ( P2 @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B2 )
         => ( P2 @ B2 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B2 )
                 => ( ! [X3: nat] :
                        ( ( member_nat2 @ X3 @ A7 )
                       => ( P2 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A7 ) ) ) ) )
         => ( P2 @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5629_finite__induct__select,axiom,
    ! [S3: set_complex,P2: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [T5: set_complex] :
              ( ( ord_less_set_complex @ T5 @ S3 )
             => ( ( P2 @ T5 )
               => ? [X3: complex] :
                    ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ S3 @ T5 ) )
                    & ( P2 @ ( insert_complex @ X3 @ T5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5630_finite__induct__select,axiom,
    ! [S3: set_Extended_enat,P2: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( P2 @ bot_bo7653980558646680370d_enat )
       => ( ! [T5: set_Extended_enat] :
              ( ( ord_le2529575680413868914d_enat @ T5 @ S3 )
             => ( ( P2 @ T5 )
               => ? [X3: extended_enat] :
                    ( ( member_Extended_enat @ X3 @ ( minus_925952699566721837d_enat @ S3 @ T5 ) )
                    & ( P2 @ ( insert_Extended_enat @ X3 @ T5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5631_finite__induct__select,axiom,
    ! [S3: set_real,P2: set_real > $o] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [T5: set_real] :
              ( ( ord_less_set_real @ T5 @ S3 )
             => ( ( P2 @ T5 )
               => ? [X3: real] :
                    ( ( member_real2 @ X3 @ ( minus_minus_set_real @ S3 @ T5 ) )
                    & ( P2 @ ( insert_real2 @ X3 @ T5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5632_finite__induct__select,axiom,
    ! [S3: set_o,P2: set_o > $o] :
      ( ( finite_finite_o @ S3 )
     => ( ( P2 @ bot_bot_set_o )
       => ( ! [T5: set_o] :
              ( ( ord_less_set_o @ T5 @ S3 )
             => ( ( P2 @ T5 )
               => ? [X3: $o] :
                    ( ( member_o2 @ X3 @ ( minus_minus_set_o @ S3 @ T5 ) )
                    & ( P2 @ ( insert_o2 @ X3 @ T5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5633_finite__induct__select,axiom,
    ! [S3: set_int,P2: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [T5: set_int] :
              ( ( ord_less_set_int @ T5 @ S3 )
             => ( ( P2 @ T5 )
               => ? [X3: int] :
                    ( ( member_int2 @ X3 @ ( minus_minus_set_int @ S3 @ T5 ) )
                    & ( P2 @ ( insert_int2 @ X3 @ T5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5634_finite__induct__select,axiom,
    ! [S3: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [T5: set_nat] :
              ( ( ord_less_set_nat @ T5 @ S3 )
             => ( ( P2 @ T5 )
               => ? [X3: nat] :
                    ( ( member_nat2 @ X3 @ ( minus_minus_set_nat @ S3 @ T5 ) )
                    & ( P2 @ ( insert_nat2 @ X3 @ T5 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5635_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat2 @ X @ B2 ) )
      = ( ( ( member_set_nat2 @ X @ B2 )
         => ( ord_less_set_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_set_nat2 @ X @ B2 )
         => ( ( ( member_set_nat2 @ X @ A2 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat2 @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
            & ( ~ ( member_set_nat2 @ X @ A2 )
             => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5636_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real2 @ X @ B2 ) )
      = ( ( ( member_real2 @ X @ B2 )
         => ( ord_less_set_real @ A2 @ B2 ) )
        & ( ~ ( member_real2 @ X @ B2 )
         => ( ( ( member_real2 @ X @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) @ B2 ) )
            & ( ~ ( member_real2 @ X @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5637_psubset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_set_o @ A2 @ ( insert_o2 @ X @ B2 ) )
      = ( ( ( member_o2 @ X @ B2 )
         => ( ord_less_set_o @ A2 @ B2 ) )
        & ( ~ ( member_o2 @ X @ B2 )
         => ( ( ( member_o2 @ X @ A2 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) @ B2 ) )
            & ( ~ ( member_o2 @ X @ A2 )
             => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5638_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int2 @ X @ B2 ) )
      = ( ( ( member_int2 @ X @ B2 )
         => ( ord_less_set_int @ A2 @ B2 ) )
        & ( ~ ( member_int2 @ X @ B2 )
         => ( ( ( member_int2 @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) ) @ B2 ) )
            & ( ~ ( member_int2 @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5639_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat2 @ X @ B2 ) )
      = ( ( ( member_nat2 @ X @ B2 )
         => ( ord_less_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_nat2 @ X @ B2 )
         => ( ( ( member_nat2 @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B2 ) )
            & ( ~ ( member_nat2 @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5640_set__replicate__Suc,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X ) )
      = ( insert_VEBT_VEBT2 @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_5641_set__replicate__Suc,axiom,
    ! [N2: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X ) )
      = ( insert_real2 @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_5642_set__replicate__Suc,axiom,
    ! [N2: nat,X: $o] :
      ( ( set_o2 @ ( replicate_o @ ( suc @ N2 ) @ X ) )
      = ( insert_o2 @ X @ bot_bot_set_o ) ) ).

% set_replicate_Suc
thf(fact_5643_set__replicate__Suc,axiom,
    ! [N2: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X ) )
      = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_5644_set__replicate__Suc,axiom,
    ! [N2: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X ) )
      = ( insert_int2 @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_5645_set__replicate__conv__if,axiom,
    ! [N2: nat,X: vEBT_VEBT] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X ) )
          = ( insert_VEBT_VEBT2 @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5646_set__replicate__conv__if,axiom,
    ! [N2: nat,X: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
          = bot_bot_set_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X ) )
          = ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5647_set__replicate__conv__if,axiom,
    ! [N2: nat,X: $o] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N2 @ X ) )
          = bot_bot_set_o ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N2 @ X ) )
          = ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5648_set__replicate__conv__if,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X ) )
          = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5649_set__replicate__conv__if,axiom,
    ! [N2: nat,X: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
          = bot_bot_set_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X ) )
          = ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5650_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5651_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ M )
          @ ( groups7108830773950497114d_enat
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5652_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5653_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5654_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( minus_minus_int @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_5655_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( minus_minus_real @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_5656_exp__eq__0__imp__not__bit,axiom,
    ! [N2: nat,A: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
        = zero_zero_int )
     => ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_5657_exp__eq__0__imp__not__bit,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        = zero_zero_nat )
     => ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_5658_bit__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 ) ) ).

% bit_Suc
thf(fact_5659_bit__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 ) ) ).

% bit_Suc
thf(fact_5660_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > extended_enat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real2 @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups2800946370649118462d_enat @ F @ A2 ) @ ( groups2800946370649118462d_enat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5661_sum__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > extended_enat] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B4: $o] :
              ( ( member_o2 @ B4 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups7198740251461348360d_enat @ F @ A2 ) @ ( groups7198740251461348360d_enat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5662_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups1752964319039525884d_enat @ F @ A2 ) @ ( groups1752964319039525884d_enat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5663_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > extended_enat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B4: int] :
              ( ( member_int2 @ B4 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups4225252721152677374d_enat @ F @ A2 ) @ ( groups4225252721152677374d_enat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5664_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
             => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ B4 ) ) )
         => ( ord_le2932123472753598470d_enat @ ( groups2433450451889696826d_enat @ F @ A2 ) @ ( groups2433450451889696826d_enat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5665_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real2 @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5666_sum__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > real] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B4: $o] :
              ( ( member_o2 @ B4 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5667_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5668_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B4: int] :
              ( ( member_int2 @ B4 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5669_sum__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_5670_ceiling__correct,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
      & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% ceiling_correct
thf(fact_5671_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5672_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5673_ceiling__split,axiom,
    ! [P2: int > $o,T: real] :
      ( ( P2 @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I5: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I5 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I5 ) ) )
           => ( P2 @ I5 ) ) ) ) ).

% ceiling_split
thf(fact_5674_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5675_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_5676_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5677_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ N3 @ M4 )
             => ( ( bit_se1146084159140164899it_int @ K @ M4 )
                = ( 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_5678_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5679_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5680_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5681_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5682_sum__count__set,axiom,
    ! [Xs: list_complex,X7: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X7 )
     => ( ( finite3207457112153483333omplex @ X7 )
       => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X7 )
          = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5683_sum__count__set,axiom,
    ! [Xs: list_Extended_enat,X7: set_Extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs ) @ X7 )
     => ( ( finite4001608067531595151d_enat @ X7 )
       => ( ( groups2027974829824023292at_nat @ ( count_101369445342291426d_enat @ Xs ) @ X7 )
          = ( size_s3941691890525107288d_enat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5684_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X7: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X7 )
     => ( ( finite5795047828879050333T_VEBT @ X7 )
       => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X7 )
          = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5685_sum__count__set,axiom,
    ! [Xs: list_int,X7: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X7 )
     => ( ( finite_finite_int @ X7 )
       => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X7 )
          = ( size_size_list_int @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5686_sum__count__set,axiom,
    ! [Xs: list_nat,X7: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X7 )
     => ( ( finite_finite_nat @ X7 )
       => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X7 )
          = ( size_size_list_nat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_5687_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real2 @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real2 @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5688_sum__strict__mono2,axiom,
    ! [B2: set_o,A2: set_o,B: $o,F: $o > real] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ( member_o2 @ B @ ( minus_minus_set_o @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: $o] :
                  ( ( member_o2 @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5689_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5690_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int2 @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int2 @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5691_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,B: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A2 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5692_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real2 @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real2 @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5693_sum__strict__mono2,axiom,
    ! [B2: set_o,A2: set_o,B: $o,F: $o > nat] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ( member_o2 @ B @ ( minus_minus_set_o @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: $o] :
                  ( ( member_o2 @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( groups8507830703676809646_o_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5694_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5695_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int2 @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int2 @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5696_sum__strict__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,B: extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: extended_enat] :
                  ( ( member_Extended_enat @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A2 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5697_ceiling__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_5698_Suc__mask__eq__exp,axiom,
    ! [N2: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_mask_eq_exp
thf(fact_5699_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_5700_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_5701_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_5702_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_5703_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_5704_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_5705_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_5706_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_5707_even__bit__succ__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5708_even__bit__succ__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_5709_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N2: nat] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ A @ N2 )
        = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5710_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_5711_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z: nat] :
      ( ~ ( member_nat2 @ N2 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
        = ( insert_nat2 @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_5712_ceiling__divide__lower,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_5713_ceiling__eq,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_5714_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_5715_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_5716_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5717_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5718_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5719_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5720_sum_Oin__pairs,axiom,
    ! [G: nat > extended_enat,M: nat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7108830773950497114d_enat
        @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5721_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5722_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_5723_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5724_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_5725_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A3: int,N: nat] :
          ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5726_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A3: nat,N: nat] :
          ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_5727_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_5728_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_5729_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_5730_sum__gp,axiom,
    ! [N2: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5731_sum__gp,axiom,
    ! [N2: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5732_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_5733_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_5734_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_5735_sum__gp__offset,axiom,
    ! [X: real,M: nat,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_5736_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5737_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5738_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5739_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5740_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_5741_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I5 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_5742_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I5 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_5743_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_5744_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_5745_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_5746_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum
thf(fact_5747_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_5748_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_5749_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_5750_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_5751_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_5752_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_5753_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_5754_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_5755_negative__eq__positive,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_5756_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_5757_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_5758_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_5759_int__dvd__int__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% int_dvd_int_iff
thf(fact_5760_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5761_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4216267220026989637d_enat @ M )
        = zero_z5237406670263579293d_enat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_5762_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_5763_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_5764_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_5765_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5766_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( semiri4216267220026989637d_enat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5767_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5768_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5769_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_5770_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_5771_of__nat__0,axiom,
    ( ( semiri4216267220026989637d_enat @ zero_zero_nat )
    = zero_z5237406670263579293d_enat ) ).

% of_nat_0
thf(fact_5772_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_5773_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_5774_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_5775_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5776_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5777_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5778_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_5779_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_5780_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_5781_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_5782_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W2: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W2 ) )
      = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W2 ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_5783_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,N2: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_5784_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).

% of_nat_add
thf(fact_5785_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_add
thf(fact_5786_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_5787_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_5788_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5789_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( times_times_nat @ M @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5790_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5791_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5792_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_5793_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_5794_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_5795_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_5796_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_5797_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5798_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5799_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5800_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_5801_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5802_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ N2 )
        = one_one_real )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5803_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5804_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_5805_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_5806_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n3304061248610475627l_real @ P2 ) ) ).

% of_nat_of_bool
thf(fact_5807_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% of_nat_of_bool
thf(fact_5808_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% of_nat_of_bool
thf(fact_5809_of__nat__sum,axiom,
    ! [F: complex > nat,A2: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5810_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5811_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5812_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_5813_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( semiri4216267220026989637d_enat @ M ) @ zero_z5237406670263579293d_enat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_5814_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_5815_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_5816_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_5817_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_5818_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4216267220026989637d_enat @ ( suc @ M ) )
      = ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( semiri4216267220026989637d_enat @ M ) ) ) ).

% of_nat_Suc
thf(fact_5819_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_5820_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_5821_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_5822_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5823_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5824_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5825_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_5826_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5827_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5828_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_5829_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5830_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5831_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_5832_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5833_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5834_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_5835_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5836_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5837_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_5838_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5839_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5840_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_5841_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5842_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5843_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_5844_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5845_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5846_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_5847_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5848_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5849_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_5850_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5851_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5852_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_5853_int__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_5854_real__arch__simple,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_5855_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_5856_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5857_mult__of__nat__commute,axiom,
    ! [X: nat,Y: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ X ) @ Y )
      = ( times_7803423173614009249d_enat @ Y @ ( semiri4216267220026989637d_enat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5858_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5859_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5860_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_5861_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% nat_less_real_le
thf(fact_5862_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N3: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% int_cases2
thf(fact_5863_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N3: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_diff_cases
thf(fact_5864_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5865_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5866_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_5867_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_5868_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5869_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5870_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5871_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_5872_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ zero_z5237406670263579293d_enat ) ).

% of_nat_less_0_iff
thf(fact_5873_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_5874_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_5875_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_5876_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_5877_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( suc @ N2 ) )
     != zero_z5237406670263579293d_enat ) ).

% of_nat_neq_0
thf(fact_5878_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_5879_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_5880_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_5881_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5882_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5883_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5884_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_5885_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5886_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5887_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5888_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_5889_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_5890_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_5891_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_5892_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_5893_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_5894_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_5895_int__cases,axiom,
    ! [Z: int] :
      ( ! [N3: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% int_cases
thf(fact_5896_int__of__nat__induct,axiom,
    ! [P2: int > $o,Z: int] :
      ( ! [N3: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ! [N3: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
       => ( P2 @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_5897_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_5898_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_5899_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_5900_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_5901_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_5902_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_5903_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_5904_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z6: int] :
        ? [N: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_5905_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_5906_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y ) ) ) ).

% of_nat_max
thf(fact_5907_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_5908_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_5909_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_5910_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_less_as_int
thf(fact_5911_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_leq_as_int
thf(fact_5912_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_5913_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5914_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_5915_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_5916_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_5917_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M3 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_5918_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_5919_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_5920_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_5921_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_5922_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z6: int] :
        ? [N: nat] :
          ( Z6
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_5923_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_5924_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_5925_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I5: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I5 ) @ bot_bot_set_int @ ( insert_int2 @ I5 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_5926_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_5927_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_5928_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5929_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5930_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_5931_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_5932_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_5933_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_5934_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_5935_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N3: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_5936_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_5937_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_5938_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_5939_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_5940_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_5941_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_5942_zdiff__int__split,axiom,
    ! [P2: int > $o,X: nat,Y: nat] :
      ( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P2 @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_5943_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5944_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5945_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5946_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5947_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5948_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M2: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5949_of__nat__code__if,axiom,
    ( semiri4216267220026989637d_enat
    = ( ^ [N: nat] :
          ( if_Extended_enat @ ( N = zero_zero_nat ) @ zero_z5237406670263579293d_enat
          @ ( produc2676513652042109336d_enat
            @ ^ [M2: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ one_on7984719198319812577d_enat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5950_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M2: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5951_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M2: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5952_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M2: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_5953_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P6: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q5 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_5954_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P6: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_5955_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5956_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5957_pochhammer__double,axiom,
    ! [Z: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5958_pochhammer__double,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5959_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I5: complex] : ( plus_plus_complex @ I5 @ one_one_complex )
          @ N
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_5960_of__nat__code,axiom,
    ( semiri4216267220026989637d_enat
    = ( ^ [N: nat] :
          ( semiri8563196900006977889d_enat
          @ ^ [I5: extended_enat] : ( plus_p3455044024723400733d_enat @ I5 @ one_on7984719198319812577d_enat )
          @ N
          @ zero_z5237406670263579293d_enat ) ) ) ).

% of_nat_code
thf(fact_5961_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I5: real] : ( plus_plus_real @ I5 @ one_one_real )
          @ N
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_5962_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I5: int] : ( plus_plus_int @ I5 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_5963_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I5: nat] : ( plus_plus_nat @ I5 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_5964_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int2 @ Xa2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int2 @ Xa2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_5965_lessThan__iff,axiom,
    ! [I: $o,K: $o] :
      ( ( member_o2 @ I @ ( set_ord_lessThan_o @ K ) )
      = ( ord_less_o @ I @ K ) ) ).

% lessThan_iff
thf(fact_5966_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat2 @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_5967_lessThan__iff,axiom,
    ! [I: extended_enat,K: extended_enat] :
      ( ( member_Extended_enat @ I @ ( set_or8419480210114673929d_enat @ K ) )
      = ( ord_le72135733267957522d_enat @ I @ K ) ) ).

% lessThan_iff
thf(fact_5968_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real2 @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_5969_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int2 @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_5970_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat2 @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_5971_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_5972_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_5973_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_5974_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_5975_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_5976_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_5977_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_5978_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_5979_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_5980_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_5981_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_5982_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_5983_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_5984_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_5985_single__Diff__lessThan,axiom,
    ! [K: real] :
      ( ( minus_minus_set_real @ ( insert_real2 @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
      = ( insert_real2 @ K @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_5986_single__Diff__lessThan,axiom,
    ! [K: $o] :
      ( ( minus_minus_set_o @ ( insert_o2 @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
      = ( insert_o2 @ K @ bot_bot_set_o ) ) ).

% single_Diff_lessThan
thf(fact_5987_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int2 @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int2 @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_5988_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat2 @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat2 @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_5989_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5990_sum_OlessThan__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5991_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5992_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_5993_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_5994_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_5995_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_5996_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_5997_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5998_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5999_int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% int_int_eq
thf(fact_6000_lessThan__non__empty,axiom,
    ! [X: real] :
      ( ( set_or5984915006950818249n_real @ X )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_6001_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_6002_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X2: set_nat] : ( ord_less_set_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6003_lessThan__def,axiom,
    ( set_or8419480210114673929d_enat
    = ( ^ [U2: extended_enat] :
          ( collec4429806609662206161d_enat
          @ ^ [X2: extended_enat] : ( ord_le72135733267957522d_enat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6004_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6005_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6006_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6007_Iio__eq__empty__iff,axiom,
    ! [N2: $o] :
      ( ( ( set_ord_lessThan_o @ N2 )
        = bot_bot_set_o )
      = ( N2 = bot_bot_o ) ) ).

% Iio_eq_empty_iff
thf(fact_6008_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_6009_lessThan__strict__subset__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ ( set_or8419480210114673929d_enat @ M ) @ ( set_or8419480210114673929d_enat @ N2 ) )
      = ( ord_le72135733267957522d_enat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6010_lessThan__strict__subset__iff,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
      = ( ord_less_real @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6011_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6012_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6013_pochhammer__pos,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_6014_pochhammer__pos,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_6015_pochhammer__pos,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_6016_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_6017_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ M )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_6018_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N2 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_6019_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ N2 )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_6020_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat2 @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_6021_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_6022_pochhammer__nonneg,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_6023_pochhammer__nonneg,axiom,
    ! [X: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_6024_pochhammer__nonneg,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_6025_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_6026_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_6027_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_6028_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_6029_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s3181272606743183617d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = one_on7984719198319812577d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s3181272606743183617d_enat @ zero_z5237406670263579293d_enat @ N2 )
          = zero_z5237406670263579293d_enat ) ) ) ).

% pochhammer_0_left
thf(fact_6030_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6031_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6032_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P2: nat > nat,N2: nat] :
      ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P2 @ X5 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : ( minus_minus_nat @ ( P2 @ X2 ) @ ( Q @ X2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_6033_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6034_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6035_pochhammer__rec,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6036_pochhammer__rec,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6037_pochhammer__rec,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ A @ ( comm_s3181272606743183617d_enat @ ( plus_p3455044024723400733d_enat @ A @ one_on7984719198319812577d_enat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_6038_pochhammer__Suc,axiom,
    ! [A: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6039_pochhammer__Suc,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( comm_s3181272606743183617d_enat @ A @ N2 ) @ ( plus_p3455044024723400733d_enat @ A @ ( semiri4216267220026989637d_enat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6040_pochhammer__Suc,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6041_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6042_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6043_pochhammer__rec_H,axiom,
    ! [Z: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6044_pochhammer__rec_H,axiom,
    ! [Z: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ Z @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ Z @ ( semiri4216267220026989637d_enat @ N2 ) ) @ ( comm_s3181272606743183617d_enat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6045_pochhammer__rec_H,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6046_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6047_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_6048_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6049_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_6050_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6051_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6052_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_6053_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6054_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6055_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_6056_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6057_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6058_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_6059_pochhammer__product_H,axiom,
    ! [Z: complex,N2: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6060_pochhammer__product_H,axiom,
    ! [Z: extended_enat,N2: nat,M: nat] :
      ( ( comm_s3181272606743183617d_enat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_7803423173614009249d_enat @ ( comm_s3181272606743183617d_enat @ Z @ N2 ) @ ( comm_s3181272606743183617d_enat @ ( plus_p3455044024723400733d_enat @ Z @ ( semiri4216267220026989637d_enat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6061_pochhammer__product_H,axiom,
    ! [Z: real,N2: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6062_pochhammer__product_H,axiom,
    ! [Z: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6063_pochhammer__product_H,axiom,
    ! [Z: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6064_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6065_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( G @ zero_zero_nat )
        @ ( groups7108830773950497114d_enat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6066_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6067_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6068_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6069_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6070_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6071_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6072_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6073_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6074_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s2602460028002588243omplex @ Z @ N2 )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6075_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s3181272606743183617d_enat @ Z @ N2 )
        = ( times_7803423173614009249d_enat @ ( comm_s3181272606743183617d_enat @ Z @ M ) @ ( comm_s3181272606743183617d_enat @ ( plus_p3455044024723400733d_enat @ Z @ ( semiri4216267220026989637d_enat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6076_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6077_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6078_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6079_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N2: nat] :
      ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = zero_zero_int )
      = ( ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_6080_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = zero_zero_nat )
      = ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_6081_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z @ P6 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6082_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z @ P6 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6083_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z @ P6 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6084_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) ) @ ( power_power_int @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6085_power__diff__sumr2,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) ) @ ( power_power_complex @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6086_power__diff__sumr2,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) ) @ ( power_power_real @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6087_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X @ P6 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6088_diff__power__eq__sum,axiom,
    ! [X: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X @ P6 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6089_diff__power__eq__sum,axiom,
    ! [X: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X @ P6 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6090_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6091_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6092_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6093_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > int,K5: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6094_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6095_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > real,K5: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6096_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6097_one__diff__power__eq_H,axiom,
    ! [X: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6098_one__diff__power__eq_H,axiom,
    ! [X: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6099_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ ( F @ I5 ) @ ( G @ I5 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_6100_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6101_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6102_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_6103_pochhammer__minus,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6104_pochhammer__minus,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6105_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_6106_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( member_int2 @ K2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int2 @ L2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_6107_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_6108_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_6109_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_6110_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_6111_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_6112_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_6113_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_6114_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_6115_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_6116_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_6117_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_6118_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_6119_Suc__0__and__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_6120_and__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_6121_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_nat
          @ ( ( M2 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_6122_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_6123_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6124_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_6125_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_6126_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_6127_norm__uminus__minus,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_6128_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_6129_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N2: nat,Z: real] :
      ( ( ( power_power_real @ W2 @ N2 )
        = ( power_power_real @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6130_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N2: nat,Z: complex] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = ( power_power_complex @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6131_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_6132_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_6133_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_6134_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_6135_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6136_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_6137_norm__triangle__le,axiom,
    ! [X: real,Y: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_6138_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_6139_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_6140_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_6141_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6142_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_6143_power__eq__1__iff,axiom,
    ! [W2: real,N2: nat] :
      ( ( ( power_power_real @ W2 @ N2 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W2 )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6144_power__eq__1__iff,axiom,
    ! [W2: complex,N2: nat] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W2 )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6145_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6146_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_6147_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6148_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6149_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ zero_zero_real @ M2 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_6150_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N2: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6151_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6152_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A3: complex,N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_6153_pochhammer__code,axiom,
    ( comm_s3181272606743183617d_enat
    = ( ^ [A3: extended_enat,N: nat] :
          ( if_Extended_enat @ ( N = zero_zero_nat ) @ one_on7984719198319812577d_enat
          @ ( set_fo2538466533108834004d_enat
            @ ^ [O: nat] : ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A3 @ ( semiri4216267220026989637d_enat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_on7984719198319812577d_enat ) ) ) ) ).

% pochhammer_code
thf(fact_6154_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A3: real,N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_6155_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A3: int,N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_6156_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A3: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_6157_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_6158_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) )
        @ ( 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_6159_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_6160_of__nat__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6161_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6162_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6163_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_6164_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6165_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( ring_1_of_int_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6166_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups2316167850115554303t_real
        @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6167_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_6168_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6169_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X2: int] :
              ( ( member_int2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6170_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2880970938130013265at_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6171_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups129246275422532515t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X2: nat] :
              ( ( member_nat2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6172_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups766887009212190081x_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6173_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups2316167850115554303t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X2: int] :
              ( ( member_int2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6174_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups97031904164794029t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6175_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups858564598930262913ex_int @ F @ A2 )
          = zero_zero_int )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_int ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6176_prod__zero__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( groups2878480467620962989at_int @ F @ A2 )
          = zero_zero_int )
        = ( ? [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_int ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6177_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups6464643781859351333omplex @ F @ A2 )
          = zero_zero_complex )
        = ( ? [X2: nat] :
              ( ( member_nat2 @ X2 @ A2 )
              & ( ( F @ X2 )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6178_prod_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_6179_prod_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_6180_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_6181_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_6182_prod_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty
thf(fact_6183_prod_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty
thf(fact_6184_prod_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups234877984723959775o_real @ G @ bot_bot_set_o )
      = one_one_real ) ).

% prod.empty
thf(fact_6185_prod_Oempty,axiom,
    ! [G: $o > complex] :
      ( ( groups4859619685533338977omplex @ G @ bot_bot_set_o )
      = one_one_complex ) ).

% prod.empty
thf(fact_6186_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_6187_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_6188_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_6189_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6190_prod_OlessThan__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6191_prod_OlessThan__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6192_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6193_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6194_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6195_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6196_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6197_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6198_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_on7984719198319812577d_enat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6199_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6200_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6201_prod__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6202_prod__mono,axiom,
    ! [A2: set_o,F: $o > real,G: $o > real] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( groups234877984723959775o_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6203_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat2 @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6204_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6205_prod__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6206_prod__mono,axiom,
    ! [A2: set_o,F: $o > nat,G: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( groups3504817904513533571_o_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6207_prod__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6208_prod__mono,axiom,
    ! [A2: set_real,F: real > int,G: real > int] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I3 ) )
            & ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6209_prod__mono,axiom,
    ! [A2: set_o,F: $o > int,G: $o > int] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I3 ) )
            & ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( groups3502327434004483295_o_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6210_prod__mono,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat2 @ I3 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I3 ) )
            & ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6211_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6212_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6213_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6214_prod__pos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6215_prod__pos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6216_prod__pos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6217_prod__ge__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6218_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups234877984723959775o_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6219_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6220_prod__ge__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6221_prod__ge__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6222_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6223_prod__ge__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6224_prod__ge__1,axiom,
    ! [A2: set_real,F: real > int] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6225_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6226_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6227_prod__zero,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_nat ) )
       => ( ( groups861055069439313189ex_nat @ F @ A2 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_6228_prod__zero,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ? [X3: int] :
            ( ( member_int2 @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_nat ) )
       => ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_6229_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X3: extended_enat] :
            ( ( member_Extended_enat @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_nat ) )
       => ( ( groups2880970938130013265at_nat @ F @ A2 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_6230_prod__zero,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_real ) )
       => ( ( groups129246275422532515t_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6231_prod__zero,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_real ) )
       => ( ( groups766887009212190081x_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6232_prod__zero,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ? [X3: int] :
            ( ( member_int2 @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_real ) )
       => ( ( groups2316167850115554303t_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6233_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X3: extended_enat] :
            ( ( member_Extended_enat @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_real ) )
       => ( ( groups97031904164794029t_real @ F @ A2 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6234_prod__zero,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ? [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_int ) )
       => ( ( groups858564598930262913ex_int @ F @ A2 )
          = zero_zero_int ) ) ) ).

% prod_zero
thf(fact_6235_prod__zero,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ? [X3: extended_enat] :
            ( ( member_Extended_enat @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_int ) )
       => ( ( groups2878480467620962989at_int @ F @ A2 )
          = zero_zero_int ) ) ) ).

% prod_zero
thf(fact_6236_prod__zero,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ A2 )
            & ( ( F @ X3 )
              = zero_zero_complex ) )
       => ( ( groups6464643781859351333omplex @ F @ A2 )
          = zero_zero_complex ) ) ) ).

% prod_zero
thf(fact_6237_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6238_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6239_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6240_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6241_prod__le__1,axiom,
    ! [A2: set_real,F: real > extended_enat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7973222482632965587d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6242_prod__le__1,axiom,
    ! [A2: set_o,F: $o > extended_enat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups783334030178737011d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6243_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > extended_enat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups7961826882256487087d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6244_prod__le__1,axiom,
    ! [A2: set_int,F: int > extended_enat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ X5 ) )
            & ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ one_on7984719198319812577d_enat ) ) )
     => ( ord_le2932123472753598470d_enat @ ( groups5078248829458667347d_enat @ F @ A2 ) @ one_on7984719198319812577d_enat ) ) ).

% prod_le_1
thf(fact_6245_prod__le__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6246_prod__le__1,axiom,
    ! [A2: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6247_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6248_prod__le__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6249_prod__le__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6250_prod__le__1,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6251_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6252_prod__dvd__prod__subset2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( groups3504817904513533571_o_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6253_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [A5: complex] :
              ( ( member_complex @ A5 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6254_prod__dvd__prod__subset2,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [A5: int] :
              ( ( member_int2 @ A5 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6255_prod__dvd__prod__subset2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [A5: extended_enat] :
              ( ( member_Extended_enat @ A5 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6256_prod__dvd__prod__subset2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [A5: real] :
              ( ( member_real2 @ A5 @ A2 )
             => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6257_prod__dvd__prod__subset2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > int,G: $o > int] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [A5: $o] :
              ( ( member_o2 @ A5 @ A2 )
             => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( groups3502327434004483295_o_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6258_prod__dvd__prod__subset2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [A5: complex] :
              ( ( member_complex @ A5 @ A2 )
             => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6259_prod__dvd__prod__subset2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [A5: extended_enat] :
              ( ( member_Extended_enat @ A5 @ A2 )
             => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6260_prod__dvd__prod__subset2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [A5: nat] :
              ( ( member_nat2 @ A5 @ A2 )
             => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ B2 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6261_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6262_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6263_prod__dvd__prod__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( dvd_dvd_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6264_prod__dvd__prod__subset,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6265_prod__dvd__prod__subset,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6266_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6267_prod__dvd__prod__subset,axiom,
    ! [B2: set_int,A2: set_int,F: int > int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6268_prod__dvd__prod__subset,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B2 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6269_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_6270_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I5 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_6271_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6272_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6273_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real2 @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real2 @ I3 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6274_less__1__prod2,axiom,
    ! [I6: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I6 )
     => ( ( member_o2 @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o2 @ I3 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups234877984723959775o_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6275_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat2 @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat2 @ I3 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6276_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6277_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int2 @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int2 @ I3 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6278_less__1__prod2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6279_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real2 @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real2 @ I3 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6280_less__1__prod2,axiom,
    ! [I6: set_o,I: $o,F: $o > int] :
      ( ( finite_finite_o @ I6 )
     => ( ( member_o2 @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o2 @ I3 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6281_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6282_less__1__prod2,axiom,
    ! [I6: set_Extended_enat,I: extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( member_Extended_enat @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I3: extended_enat] :
                ( ( member_Extended_enat @ I3 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups2878480467620962989at_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6283_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6284_less__1__prod,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups97031904164794029t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6285_less__1__prod,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real2 @ I3 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6286_less__1__prod,axiom,
    ! [I6: set_o,F: $o > real] :
      ( ( finite_finite_o @ I6 )
     => ( ( I6 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o2 @ I3 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups234877984723959775o_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6287_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat2 @ I3 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6288_less__1__prod,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int2 @ I3 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6289_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6290_less__1__prod,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( I6 != bot_bo7653980558646680370d_enat )
       => ( ! [I3: extended_enat] :
              ( ( member_Extended_enat @ I3 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups2878480467620962989at_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6291_less__1__prod,axiom,
    ! [I6: set_real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real2 @ I3 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6292_less__1__prod,axiom,
    ! [I6: set_o,F: $o > int] :
      ( ( finite_finite_o @ I6 )
     => ( ( I6 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o2 @ I3 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_6293_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ A2 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups861055069439313189ex_nat @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6294_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups1707563613775114915nt_nat @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6295_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > nat] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2880970938130013265at_nat @ G @ A2 )
          = ( times_times_nat @ ( groups2880970938130013265at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2880970938130013265at_nat @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6296_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ A2 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups858564598930262913ex_int @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6297_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > int] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups2878480467620962989at_int @ G @ A2 )
          = ( times_times_int @ ( groups2878480467620962989at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups2878480467620962989at_int @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6298_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups766887009212190081x_real @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6299_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups2316167850115554303t_real @ G @ A2 )
          = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups2316167850115554303t_real @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6300_prod_Osubset__diff,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,G: extended_enat > real] :
      ( ( ord_le7203529160286727270d_enat @ B2 @ A2 )
     => ( ( finite4001608067531595151d_enat @ A2 )
       => ( ( groups97031904164794029t_real @ G @ A2 )
          = ( times_times_real @ ( groups97031904164794029t_real @ G @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( groups97031904164794029t_real @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6301_prod_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > complex] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups3708469109370488835omplex @ G @ A2 )
          = ( times_times_complex @ ( groups3708469109370488835omplex @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups3708469109370488835omplex @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6302_prod_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > complex] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups7440179247065528705omplex @ G @ A2 )
          = ( times_times_complex @ ( groups7440179247065528705omplex @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups7440179247065528705omplex @ G @ B2 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6303_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ T3 )
              = ( groups4696554848551431203al_nat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6304_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ T3 )
              = ( groups3504817904513533571_o_nat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6305_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups861055069439313189ex_nat @ G @ T3 )
              = ( groups861055069439313189ex_nat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6306_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1707563613775114915nt_nat @ G @ T3 )
              = ( groups1707563613775114915nt_nat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6307_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2880970938130013265at_nat @ G @ T3 )
              = ( groups2880970938130013265at_nat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6308_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4694064378042380927al_int @ G @ T3 )
              = ( groups4694064378042380927al_int @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6309_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > int,H2: $o > int] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3502327434004483295_o_int @ G @ T3 )
              = ( groups3502327434004483295_o_int @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6310_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups858564598930262913ex_int @ G @ T3 )
              = ( groups858564598930262913ex_int @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6311_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2878480467620962989at_int @ G @ T3 )
              = ( groups2878480467620962989at_int @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6312_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T3 )
              = ( groups1681761925125756287l_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_6313_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ S3 )
              = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6314_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H2: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ S3 )
              = ( groups3504817904513533571_o_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6315_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups861055069439313189ex_nat @ G @ S3 )
              = ( groups861055069439313189ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6316_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: int] :
                ( ( member_int2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1707563613775114915nt_nat @ G @ S3 )
              = ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6317_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2880970938130013265at_nat @ G @ S3 )
              = ( groups2880970938130013265at_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6318_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4694064378042380927al_int @ G @ S3 )
              = ( groups4694064378042380927al_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6319_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H2: $o > int,G: $o > int] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: $o] :
                ( ( member_o2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3502327434004483295_o_int @ G @ S3 )
              = ( groups3502327434004483295_o_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6320_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups858564598930262913ex_int @ G @ S3 )
              = ( groups858564598930262913ex_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6321_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,H2: extended_enat > int,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_int ) )
         => ( ! [X5: extended_enat] :
                ( ( member_Extended_enat @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2878480467620962989at_int @ G @ S3 )
              = ( groups2878480467620962989at_int @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6322_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real2 @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S3 )
              = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_6323_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ T3 )
            = ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6324_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ T3 )
            = ( groups1707563613775114915nt_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6325_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups2880970938130013265at_nat @ G @ T3 )
            = ( groups2880970938130013265at_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6326_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ T3 )
            = ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6327_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups2878480467620962989at_int @ G @ T3 )
            = ( groups2878480467620962989at_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6328_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T3 )
            = ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6329_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G @ T3 )
            = ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6330_prod_Omono__neutral__right,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G @ T3 )
            = ( groups97031904164794029t_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6331_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ T3 )
            = ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6332_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G @ T3 )
            = ( groups7440179247065528705omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_6333_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ S3 )
            = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6334_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ S3 )
            = ( groups1707563613775114915nt_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6335_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups2880970938130013265at_nat @ G @ S3 )
            = ( groups2880970938130013265at_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6336_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ S3 )
            = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6337_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups2878480467620962989at_int @ G @ S3 )
            = ( groups2878480467620962989at_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6338_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S3 )
            = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6339_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G @ S3 )
            = ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6340_prod_Omono__neutral__left,axiom,
    ! [T3: set_Extended_enat,S3: set_Extended_enat,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ T3 )
     => ( ( ord_le7203529160286727270d_enat @ S3 @ T3 )
       => ( ! [X5: extended_enat] :
              ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups97031904164794029t_real @ G @ S3 )
            = ( groups97031904164794029t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6341_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ S3 )
            = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6342_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G @ S3 )
            = ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_6343_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) )
               => ( ( groups4696554848551431203al_nat @ G @ A2 )
                  = ( groups4696554848551431203al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6344_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ C4 )
                  = ( groups3504817904513533571_o_nat @ H2 @ C4 ) )
               => ( ( groups3504817904513533571_o_nat @ G @ A2 )
                  = ( groups3504817904513533571_o_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6345_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups861055069439313189ex_nat @ G @ C4 )
                  = ( groups861055069439313189ex_nat @ H2 @ C4 ) )
               => ( ( groups861055069439313189ex_nat @ G @ A2 )
                  = ( groups861055069439313189ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6346_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups1707563613775114915nt_nat @ G @ C4 )
                  = ( groups1707563613775114915nt_nat @ H2 @ C4 ) )
               => ( ( groups1707563613775114915nt_nat @ G @ A2 )
                  = ( groups1707563613775114915nt_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6347_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups2880970938130013265at_nat @ G @ C4 )
                  = ( groups2880970938130013265at_nat @ H2 @ C4 ) )
               => ( ( groups2880970938130013265at_nat @ G @ A2 )
                  = ( groups2880970938130013265at_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6348_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups4694064378042380927al_int @ G @ C4 )
                  = ( groups4694064378042380927al_int @ H2 @ C4 ) )
               => ( ( groups4694064378042380927al_int @ G @ A2 )
                  = ( groups4694064378042380927al_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6349_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > int,H2: $o > int] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups3502327434004483295_o_int @ G @ C4 )
                  = ( groups3502327434004483295_o_int @ H2 @ C4 ) )
               => ( ( groups3502327434004483295_o_int @ G @ A2 )
                  = ( groups3502327434004483295_o_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6350_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups858564598930262913ex_int @ G @ C4 )
                  = ( groups858564598930262913ex_int @ H2 @ C4 ) )
               => ( ( groups858564598930262913ex_int @ G @ A2 )
                  = ( groups858564598930262913ex_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6351_prod_Osame__carrierI,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups2878480467620962989at_int @ G @ C4 )
                  = ( groups2878480467620962989at_int @ H2 @ C4 ) )
               => ( ( groups2878480467620962989at_int @ G @ A2 )
                  = ( groups2878480467620962989at_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6352_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_real ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_6353_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ A2 )
                  = ( groups4696554848551431203al_nat @ H2 @ B2 ) )
                = ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6354_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ A2 )
                  = ( groups3504817904513533571_o_nat @ H2 @ B2 ) )
                = ( ( groups3504817904513533571_o_nat @ G @ C4 )
                  = ( groups3504817904513533571_o_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6355_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups861055069439313189ex_nat @ G @ A2 )
                  = ( groups861055069439313189ex_nat @ H2 @ B2 ) )
                = ( ( groups861055069439313189ex_nat @ G @ C4 )
                  = ( groups861055069439313189ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6356_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B2 @ C4 )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: int] :
                  ( ( member_int2 @ B4 @ ( minus_minus_set_int @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups1707563613775114915nt_nat @ G @ A2 )
                  = ( groups1707563613775114915nt_nat @ H2 @ B2 ) )
                = ( ( groups1707563613775114915nt_nat @ G @ C4 )
                  = ( groups1707563613775114915nt_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6357_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_nat ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_nat ) )
             => ( ( ( groups2880970938130013265at_nat @ G @ A2 )
                  = ( groups2880970938130013265at_nat @ H2 @ B2 ) )
                = ( ( groups2880970938130013265at_nat @ G @ C4 )
                  = ( groups2880970938130013265at_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6358_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups4694064378042380927al_int @ G @ A2 )
                  = ( groups4694064378042380927al_int @ H2 @ B2 ) )
                = ( ( groups4694064378042380927al_int @ G @ C4 )
                  = ( groups4694064378042380927al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6359_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B2: set_o,G: $o > int,H2: $o > int] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B2 @ C4 )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: $o] :
                  ( ( member_o2 @ B4 @ ( minus_minus_set_o @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups3502327434004483295_o_int @ G @ A2 )
                  = ( groups3502327434004483295_o_int @ H2 @ B2 ) )
                = ( ( groups3502327434004483295_o_int @ G @ C4 )
                  = ( groups3502327434004483295_o_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6360_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups858564598930262913ex_int @ G @ A2 )
                  = ( groups858564598930262913ex_int @ H2 @ B2 ) )
                = ( ( groups858564598930262913ex_int @ G @ C4 )
                  = ( groups858564598930262913ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6361_prod_Osame__carrier,axiom,
    ! [C4: set_Extended_enat,A2: set_Extended_enat,B2: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ C4 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ C4 )
       => ( ( ord_le7203529160286727270d_enat @ B2 @ C4 )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ ( minus_925952699566721837d_enat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_int ) )
           => ( ! [B4: extended_enat] :
                  ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_int ) )
             => ( ( ( groups2878480467620962989at_int @ G @ A2 )
                  = ( groups2878480467620962989at_int @ H2 @ B2 ) )
                = ( ( groups2878480467620962989at_int @ G @ C4 )
                  = ( groups2878480467620962989at_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6362_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = one_one_real ) )
           => ( ! [B4: real] :
                  ( ( member_real2 @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B2 ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_6363_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6364_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6365_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6366_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6367_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6368_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6369_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6370_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6371_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( G @ M ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6372_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_7803423173614009249d_enat @ ( G @ M ) @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6373_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6374_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6375_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6376_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_complex @ ( G @ ( suc @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6377_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_7803423173614009249d_enat @ ( G @ ( suc @ N2 ) ) @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6378_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6379_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6380_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6381_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6382_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( G @ zero_zero_nat )
        @ ( groups7961826882256487087d_enat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6383_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6384_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6385_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_real @ ( G @ M )
          @ ( groups129246275422532515t_real
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6386_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_complex @ ( G @ M )
          @ ( groups6464643781859351333omplex
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6387_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_7803423173614009249d_enat @ ( G @ M )
          @ ( groups7961826882256487087d_enat
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6388_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6389_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6390_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_6391_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_6392_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6393_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > real,G: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [I3: extended_enat] :
            ( ( member_Extended_enat @ I3 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( ord_less_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6394_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6395_prod__mono__strict,axiom,
    ! [A2: set_o,F: $o > real,G: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_o )
         => ( ord_less_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( groups234877984723959775o_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6396_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I3: nat] :
            ( ( member_nat2 @ I3 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6397_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6398_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6399_prod__mono__strict,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ! [I3: extended_enat] :
            ( ( member_Extended_enat @ I3 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bo7653980558646680370d_enat )
         => ( ord_less_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( groups2880970938130013265at_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6400_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6401_prod__mono__strict,axiom,
    ! [A2: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ A2 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A2 != bot_bot_set_o )
         => ( ord_less_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( groups3504817904513533571_o_nat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6402_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6403_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2880970938130013265at_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups2880970938130013265at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6404_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real2 @ X @ A2 )
       => ( ( groups4696554848551431203al_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups4696554848551431203al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6405_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o2 @ X @ A2 )
       => ( ( groups3504817904513533571_o_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups3504817904513533571_o_nat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6406_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int2 @ X @ A2 )
       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
          = ( times_times_nat @ ( G @ X ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6407_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6408_prod_Oremove,axiom,
    ! [A2: set_Extended_enat,X: extended_enat,G: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( member_Extended_enat @ X @ A2 )
       => ( ( groups2878480467620962989at_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups2878480467620962989at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6409_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real2 @ X @ A2 )
       => ( ( groups4694064378042380927al_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups4694064378042380927al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6410_prod_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o2 @ X @ A2 )
       => ( ( groups3502327434004483295_o_int @ G @ A2 )
          = ( times_times_int @ ( G @ X ) @ ( groups3502327434004483295_o_int @ G @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6411_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6412_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups861055069439313189ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6413_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > nat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2880970938130013265at_nat @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups2880970938130013265at_nat @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6414_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4696554848551431203al_nat @ G @ ( insert_real2 @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups4696554848551431203al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6415_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > nat,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups3504817904513533571_o_nat @ G @ ( insert_o2 @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups3504817904513533571_o_nat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6416_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > nat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G @ ( insert_int2 @ X @ A2 ) )
        = ( times_times_nat @ ( G @ X ) @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6417_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups858564598930262913ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6418_prod_Oinsert__remove,axiom,
    ! [A2: set_Extended_enat,G: extended_enat > int,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( groups2878480467620962989at_int @ G @ ( insert_Extended_enat @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups2878480467620962989at_int @ G @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ X @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6419_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > int,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups4694064378042380927al_int @ G @ ( insert_real2 @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups4694064378042380927al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6420_prod_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > int,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups3502327434004483295_o_int @ G @ ( insert_o2 @ X @ A2 ) )
        = ( times_times_int @ ( G @ X ) @ ( groups3502327434004483295_o_int @ G @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6421_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6422_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6423_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > complex,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6424_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6425_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6426_prod_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6427_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups861055069439313189ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups861055069439313189ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6428_prod_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > nat,C: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups2880970938130013265at_nat @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2880970938130013265at_nat
              @ ^ [K2: extended_enat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2880970938130013265at_nat @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6429_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups4696554848551431203al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups4696554848551431203al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups4696554848551431203al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6430_prod_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > nat,C: $o > nat] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o2 @ A @ S3 )
         => ( ( groups3504817904513533571_o_nat
              @ ^ [K2: $o] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups3504817904513533571_o_nat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o2 @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o2 @ A @ S3 )
         => ( ( groups3504817904513533571_o_nat
              @ ^ [K2: $o] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups3504817904513533571_o_nat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o2 @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6431_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > nat,C: int > nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int2 @ A @ S3 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups1707563613775114915nt_nat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int2 @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int2 @ A @ S3 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1707563613775114915nt_nat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int2 @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6432_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_int @ ( B @ A ) @ ( groups858564598930262913ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups858564598930262913ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups858564598930262913ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6433_prod_Odelta__remove,axiom,
    ! [S3: set_Extended_enat,A: extended_enat,B: extended_enat > int,C: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( ( member_Extended_enat @ A @ S3 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_int @ ( B @ A ) @ ( groups2878480467620962989at_int @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
        & ( ~ ( member_Extended_enat @ A @ S3 )
         => ( ( groups2878480467620962989at_int
              @ ^ [K2: extended_enat] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2878480467620962989at_int @ C @ ( minus_925952699566721837d_enat @ S3 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6434_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > int,C: real > int] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real2 @ A @ S3 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_int @ ( B @ A ) @ ( groups4694064378042380927al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real2 @ A @ S3 )
         => ( ( groups4694064378042380927al_int
              @ ^ [K2: real] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups4694064378042380927al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real2 @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6435_prod_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > int,C: $o > int] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o2 @ A @ S3 )
         => ( ( groups3502327434004483295_o_int
              @ ^ [K2: $o] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_int @ ( B @ A ) @ ( groups3502327434004483295_o_int @ C @ ( minus_minus_set_o @ S3 @ ( insert_o2 @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o2 @ A @ S3 )
         => ( ( groups3502327434004483295_o_int
              @ ^ [K2: $o] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups3502327434004483295_o_int @ C @ ( minus_minus_set_o @ S3 @ ( insert_o2 @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6436_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6437_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F5: nat > nat > nat,A3: nat,B3: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A3 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B3 @ ( F5 @ A3 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_6438_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa2 )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa2 )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_6439_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real2 @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6440_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > real] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B4: $o] :
              ( ( member_o2 @ B4 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A2 ) @ ( groups234877984723959775o_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6441_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6442_prod__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B4: int] :
              ( ( member_int2 @ B4 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A5: int] :
                ( ( member_int2 @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6443_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > real] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups97031904164794029t_real @ F @ A2 ) @ ( groups97031904164794029t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6444_prod__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real2 @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A5: real] :
                ( ( member_real2 @ A5 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A5 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6445_prod__mono2,axiom,
    ! [B2: set_o,A2: set_o,F: $o > int] :
      ( ( finite_finite_o @ B2 )
     => ( ( ord_less_eq_set_o @ A2 @ B2 )
       => ( ! [B4: $o] :
              ( ( member_o2 @ B4 @ ( minus_minus_set_o @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A5: $o] :
                ( ( member_o2 @ A5 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A5 ) ) )
           => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( groups3502327434004483295_o_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6446_prod__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A5 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6447_prod__mono2,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat,F: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ! [B4: extended_enat] :
              ( ( member_Extended_enat @ B4 @ ( minus_925952699566721837d_enat @ B2 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B4 ) ) )
         => ( ! [A5: extended_enat] :
                ( ( member_Extended_enat @ A5 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A5 ) ) )
           => ( ord_less_eq_int @ ( groups2878480467620962989at_int @ F @ A2 ) @ ( groups2878480467620962989at_int @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6448_prod__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B4: nat] :
              ( ( member_nat2 @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B4 ) ) )
         => ( ! [A5: nat] :
                ( ( member_nat2 @ A5 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A5 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B2 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6449_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide1717551699836669952omplex @ ( groups3708469109370488835omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups3708469109370488835omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6450_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > complex,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_Extended_enat @ A @ A2 )
           => ( ( groups4622424608036095791omplex @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( divide1717551699836669952omplex @ ( groups4622424608036095791omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Extended_enat @ A @ A2 )
           => ( ( groups4622424608036095791omplex @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups4622424608036095791omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6451_prod__diff1,axiom,
    ! [A2: set_real,F: real > complex,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_real2 @ A @ A2 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
              = ( divide1717551699836669952omplex @ ( groups713298508707869441omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real2 @ A @ A2 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
              = ( groups713298508707869441omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6452_prod__diff1,axiom,
    ! [A2: set_o,F: $o > complex,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_o2 @ A @ A2 )
           => ( ( groups4859619685533338977omplex @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
              = ( divide1717551699836669952omplex @ ( groups4859619685533338977omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_o2 @ A @ A2 )
           => ( ( groups4859619685533338977omplex @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
              = ( groups4859619685533338977omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6453_prod__diff1,axiom,
    ! [A2: set_int,F: int > complex,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_int2 @ A @ A2 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
              = ( divide1717551699836669952omplex @ ( groups7440179247065528705omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int2 @ A @ A2 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A2 @ ( insert_int2 @ A @ bot_bot_set_int ) ) )
              = ( groups7440179247065528705omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6454_prod__diff1,axiom,
    ! [A2: set_nat,F: nat > complex,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_nat2 @ A @ A2 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
              = ( divide1717551699836669952omplex @ ( groups6464643781859351333omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_nat2 @ A @ A2 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
              = ( groups6464643781859351333omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6455_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > nat,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups861055069439313189ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups861055069439313189ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6456_prod__diff1,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat,A: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_Extended_enat @ A @ A2 )
           => ( ( groups2880970938130013265at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( divide_divide_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Extended_enat @ A @ A2 )
           => ( ( groups2880970938130013265at_nat @ F @ ( minus_925952699566721837d_enat @ A2 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups2880970938130013265at_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6457_prod__diff1,axiom,
    ! [A2: set_real,F: real > nat,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_real2 @ A @ A2 )
           => ( ( groups4696554848551431203al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
              = ( divide_divide_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real2 @ A @ A2 )
           => ( ( groups4696554848551431203al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real2 @ A @ bot_bot_set_real ) ) )
              = ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6458_prod__diff1,axiom,
    ! [A2: set_o,F: $o > nat,A: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( F @ A )
         != zero_zero_nat )
       => ( ( ( member_o2 @ A @ A2 )
           => ( ( groups3504817904513533571_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
              = ( divide_divide_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_o2 @ A @ A2 )
           => ( ( groups3504817904513533571_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o2 @ A @ bot_bot_set_o ) ) )
              = ( groups3504817904513533571_o_nat @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6459_log__of__power__less,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_6460_pochhammer__Suc__prod,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ A @ ( semiri4216267220026989637d_enat @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6461_pochhammer__Suc__prod,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I5: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6462_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6463_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6464_pochhammer__prod__rev,axiom,
    ( comm_s3181272606743183617d_enat
    = ( ^ [A3: extended_enat,N: nat] :
          ( groups7961826882256487087d_enat
          @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ A3 @ ( semiri4216267220026989637d_enat @ ( minus_minus_nat @ N @ I5 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6465_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A3: real,N: nat] :
          ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I5 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6466_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A3: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I5: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I5 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6467_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A3: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I5: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I5 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_6468_log__of__power__le,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_6469_prod_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I5: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6470_prod_Oin__pairs,axiom,
    ! [G: nat > complex,M: nat,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [I5: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6471_prod_Oin__pairs,axiom,
    ! [G: nat > extended_enat,M: nat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I5: nat] : ( times_7803423173614009249d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6472_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6473_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_6474_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A3: nat] : ( plus_plus_int @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_6475_sum__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A3: nat] : ( plus_plus_complex @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_6476_sum__atLeastAtMost__code,axiom,
    ! [F: nat > extended_enat,A: nat,B: nat] :
      ( ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2538466533108834004d_enat
        @ ^ [A3: nat] : ( plus_p3455044024723400733d_enat @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_z5237406670263579293d_enat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6477_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A3: nat] : ( plus_plus_nat @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6478_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A3: nat] : ( plus_plus_real @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_6479_pochhammer__Suc__prod__rev,axiom,
    ! [A: extended_enat,N2: nat] :
      ( ( comm_s3181272606743183617d_enat @ A @ ( suc @ N2 ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ A @ ( semiri4216267220026989637d_enat @ ( minus_minus_nat @ N2 @ I5 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6480_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
      = ( groups129246275422532515t_real
        @ ^ [I5: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I5 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6481_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I5 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6482_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I5 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6483_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_6484_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_6485_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_6486_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_6487_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_6488_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_6489_power__half__series,axiom,
    ( sums_real
    @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_6490_sums__zero,axiom,
    ( sums_nat
    @ ^ [N: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_6491_sums__zero,axiom,
    ( sums_real
    @ ^ [N: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_6492_sums__zero,axiom,
    ( sums_int
    @ ^ [N: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_6493_sums__zero,axiom,
    ( sums_complex
    @ ^ [N: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_6494_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A2: set_nat,S5: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S5
            = ( plus_plus_real @ S3
              @ ( groups6591440286371151544t_real
                @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
                @ A2 ) ) )
         => ( sums_real
            @ ^ [N: nat] : ( if_real @ ( member_nat2 @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_6495_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_6496_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N2: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_6497_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_6498_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ! [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6499_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
        = ( ! [X2: int] :
              ( ( member_int2 @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6500_prod__pos__nat__iff,axiom,
    ! [A2: set_Extended_enat,F: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups2880970938130013265at_nat @ F @ A2 ) )
        = ( ! [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6501_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
        = ( ! [X2: nat] :
              ( ( member_nat2 @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6502_int__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_prod
thf(fact_6503_int__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_prod
thf(fact_6504_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_6505_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6506_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6507_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6508_sums__0,axiom,
    ! [F: nat > nat] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_nat )
     => ( sums_nat @ F @ zero_zero_nat ) ) ).

% sums_0
thf(fact_6509_sums__0,axiom,
    ! [F: nat > real] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_real )
     => ( sums_real @ F @ zero_zero_real ) ) ).

% sums_0
thf(fact_6510_sums__0,axiom,
    ! [F: nat > int] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_int )
     => ( sums_int @ F @ zero_zero_int ) ) ).

% sums_0
thf(fact_6511_sums__0,axiom,
    ! [F: nat > complex] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_complex )
     => ( sums_complex @ F @ zero_zero_complex ) ) ).

% sums_0
thf(fact_6512_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R4: nat] : ( if_nat @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6513_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R4: nat] : ( if_real @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6514_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R4: nat] : ( if_int @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6515_sums__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( sums_complex
      @ ^ [R4: nat] : ( if_complex @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_complex )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_6516_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6517_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6518_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6519_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6520_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6521_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6522_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6523_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6524_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6525_sums__Suc__imp,axiom,
    ! [F: nat > real,S: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S )
       => ( sums_real @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_6526_sums__Suc__imp,axiom,
    ! [F: nat > complex,S: complex] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N: nat] : ( F @ ( suc @ N ) )
          @ S )
       => ( sums_complex @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_6527_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_6528_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6529_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6530_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6531_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ( F @ I3 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N2 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_6532_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > complex,S: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ( F @ I3 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N2 ) )
          @ S )
        = ( sums_complex @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_6533_sums__finite,axiom,
    ! [N7: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_6534_sums__finite,axiom,
    ! [N7: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_6535_sums__finite,axiom,
    ! [N7: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_6536_sums__finite,axiom,
    ! [N7: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_6537_sums__If__finite,axiom,
    ! [P2: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_int
        @ ^ [R4: nat] : ( if_int @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6538_sums__If__finite,axiom,
    ! [P2: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_complex
        @ ^ [R4: nat] : ( if_complex @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6539_sums__If__finite,axiom,
    ! [P2: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_nat
        @ ^ [R4: nat] : ( if_nat @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6540_sums__If__finite,axiom,
    ! [P2: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_real
        @ ^ [R4: nat] : ( if_real @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6541_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_int
        @ ^ [R4: nat] : ( if_int @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6542_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_complex
        @ ^ [R4: nat] : ( if_complex @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6543_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_nat
        @ ^ [R4: nat] : ( if_nat @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6544_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_real
        @ ^ [R4: nat] : ( if_real @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_6545_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_6546_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_6547_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_6548_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_6549_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_6550_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_6551_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int2 @ Xa2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int2 @ Xa2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_6552_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int2 @ K @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int2 @ L @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_int2 @ K @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int2 @ L @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_6553_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( 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_6554_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_6555_powr__0,axiom,
    ! [Z: real] :
      ( ( powr_real @ zero_zero_real @ Z )
      = zero_zero_real ) ).

% powr_0
thf(fact_6556_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_6557_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_6558_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_6559_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_6560_suminf__zero,axiom,
    ( ( suminf_complex
      @ ^ [N: nat] : zero_zero_complex )
    = zero_zero_complex ) ).

% suminf_zero
thf(fact_6561_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_6562_floor__zero,axiom,
    ( ( archim6058952711729229775r_real @ zero_zero_real )
    = zero_zero_int ) ).

% floor_zero
thf(fact_6563_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_6564_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_6565_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_6566_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_6567_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_6568_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_6569_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_6570_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_6571_zero__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_floor
thf(fact_6572_floor__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_6573_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_6574_zero__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% zero_less_floor
thf(fact_6575_floor__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_le_zero
thf(fact_6576_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_6577_one__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% one_le_floor
thf(fact_6578_floor__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_less_one
thf(fact_6579_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_6580_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_6581_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_less_floor
thf(fact_6582_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_6583_one__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_6584_floor__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_6585_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_6586_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_6587_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_6588_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_6589_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_6590_of__int__floor__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_6591_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_6592_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_6593_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_6594_le__floor__add,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_6595_floor__add__int,axiom,
    ! [X: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_6596_int__add__floor,axiom,
    ! [Z: int,X: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_6597_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_6598_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_6599_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_6600_powr__add,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).

% powr_add
thf(fact_6601_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_6602_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( ( suminf_complex @ F )
          = ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_6603_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_6604_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_6605_one__add__floor,axiom,
    ! [X: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_6606_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_6607_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_6608_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_6609_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_6610_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_6611_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
      = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_6612_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_6613_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_6614_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_6615_floor__split,axiom,
    ! [P2: int > $o,T: real] :
      ( ( P2 @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I5: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I5 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I5 ) @ one_one_real ) ) )
           => ( P2 @ I5 ) ) ) ) ).

% floor_split
thf(fact_6616_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_6617_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% less_floor_iff
thf(fact_6618_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_6619_floor__correct,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_6620_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_6621_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_6622_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_6623_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_6624_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_6625_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_6626_floor__divide__lower,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_6627_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_6628_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_6629_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_6630_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_6631_floor__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_6632_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6633_gbinomial__minus,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_6634_gbinomial__minus,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_6635_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_6636_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_6637_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K3: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
           => ( ( ~ ( ( member_int2 @ K3 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int2 @ L4 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P2 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P2 @ K3 @ L4 ) ) )
       => ( P2 @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_6638_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_6639_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_6640_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_6641_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_6642_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_6643_floor__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_6644_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_6645_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P2 @ I3 @ J2 ) ) )
       => ( P2 @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_6646_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_6647_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_6648_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_6649_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X2 ) ) @ ( archim7802044766580827645g_real @ X2 ) @ ( archim6058952711729229775r_real @ X2 ) ) ) ) ).

% round_altdef
thf(fact_6650_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6651_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6652_atMost__iff,axiom,
    ! [I: $o,K: $o] :
      ( ( member_o2 @ I @ ( set_ord_atMost_o @ K ) )
      = ( ord_less_eq_o @ I @ K ) ) ).

% atMost_iff
thf(fact_6653_atMost__iff,axiom,
    ! [I: filter_nat,K: filter_nat] :
      ( ( member_filter_nat @ I @ ( set_or9144418160755794905er_nat @ K ) )
      = ( ord_le2510731241096832064er_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_6654_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real2 @ I @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I @ K ) ) ).

% atMost_iff
thf(fact_6655_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat2 @ I @ ( set_or4236626031148496127et_nat @ K ) )
      = ( ord_less_eq_set_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_6656_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int2 @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_6657_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat2 @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_6658_atMost__subset__iff,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( ord_le2426478655948331894er_nat @ ( set_or9144418160755794905er_nat @ X ) @ ( set_or9144418160755794905er_nat @ Y ) )
      = ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6659_atMost__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ X ) @ ( set_ord_atMost_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6660_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6661_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6662_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6663_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_6664_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_6665_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_6666_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_6667_of__real__add,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_add
thf(fact_6668_of__real__add,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_add
thf(fact_6669_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_6670_Icc__subset__Iic__iff,axiom,
    ! [L: filter_nat,H2: filter_nat,H3: filter_nat] :
      ( ( ord_le2426478655948331894er_nat @ ( set_or1955772592623580779er_nat @ L @ H2 ) @ ( set_or9144418160755794905er_nat @ H3 ) )
      = ( ~ ( ord_le2510731241096832064er_nat @ L @ H2 )
        | ( ord_le2510731241096832064er_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6671_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H2: set_nat,H3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
      = ( ~ ( ord_less_eq_set_nat @ L @ H2 )
        | ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6672_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H2: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H2 )
        | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6673_Icc__subset__Iic__iff,axiom,
    ! [L: int,H2: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L @ H2 )
        | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6674_Icc__subset__Iic__iff,axiom,
    ! [L: real,H2: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L @ H2 )
        | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_6675_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6676_sum_OatMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6677_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6678_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6679_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6680_prod_OatMost__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6681_prod_OatMost__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6682_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6683_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6684_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_6685_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: real] :
      ( bot_bot_set_real
     != ( set_ord_atMost_real @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6686_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: $o] :
      ( bot_bot_set_o
     != ( set_ord_atMost_o @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6687_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: int] :
      ( bot_bot_set_int
     != ( set_ord_atMost_int @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6688_not__empty__eq__Iic__eq__empty,axiom,
    ! [H2: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atMost_nat @ H2 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6689_atMost__def,axiom,
    ( set_or9144418160755794905er_nat
    = ( ^ [U2: filter_nat] :
          ( collect_filter_nat
          @ ^ [X2: filter_nat] : ( ord_le2510731241096832064er_nat @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6690_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X2: real] : ( ord_less_eq_real @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6691_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6692_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6693_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6694_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_6695_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_6696_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat2 @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_6697_not__Iic__le__Icc,axiom,
    ! [H2: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_6698_not__Iic__le__Icc,axiom,
    ! [H2: real,L3: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_6699_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_6700_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_6701_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_6702_Iic__subset__Iio__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_or8332593352340944941d_enat @ A ) @ ( set_or8419480210114673929d_enat @ B ) )
      = ( ord_le72135733267957522d_enat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_6703_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_6704_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_6705_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_6706_norm__less__p1,axiom,
    ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_6707_norm__less__p1,axiom,
    ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_6708_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6709_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( G @ zero_zero_nat )
        @ ( groups7108830773950497114d_enat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6710_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6711_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6712_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( minus_minus_int @ ( F @ I5 ) @ ( F @ ( suc @ I5 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_6713_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( minus_minus_real @ ( F @ I5 ) @ ( F @ ( suc @ I5 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_6714_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N2: nat,D: nat > complex] :
      ( ( ! [X2: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups2073611262835488442omplex
              @ ^ [I5: nat] : ( times_times_complex @ ( D @ I5 ) @ ( power_power_complex @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_eq_nat @ I5 @ N2 )
           => ( ( C @ I5 )
              = ( D @ I5 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_6715_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N2: nat,D: nat > real] :
      ( ( ! [X2: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( D @ I5 ) @ ( power_power_real @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_eq_nat @ I5 @ N2 )
           => ( ( C @ I5 )
              = ( D @ I5 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_6716_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6717_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6718_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( G @ zero_zero_nat )
        @ ( groups7961826882256487087d_enat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6719_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6720_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6721_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( groups3542108847815614940at_nat @ ( A @ I5 ) @ ( set_ord_lessThan_nat @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_6722_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( groups6591440286371151544t_real @ ( A @ I5 ) @ ( set_ord_lessThan_nat @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_6723_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( groups705719431365010083at_int @ ( A @ I5 ) @ ( set_ord_lessThan_nat @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_6724_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( groups708209901874060359at_nat @ ( A @ I5 ) @ ( set_ord_lessThan_nat @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_6725_frac__eq,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = X )
      = ( ( ord_less_eq_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% frac_eq
thf(fact_6726_polyfun__eq__0,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( ! [X2: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_complex ) )
      = ( ! [I5: nat] :
            ( ( ord_less_eq_nat @ I5 @ N2 )
           => ( ( C @ I5 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_6727_polyfun__eq__0,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( ! [X2: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_real ) )
      = ( ! [I5: nat] :
            ( ( ord_less_eq_nat @ I5 @ N2 )
           => ( ( C @ I5 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_6728_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N2: nat,K: nat] :
      ( ! [W: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ W @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_6729_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N2: nat,K: nat] :
      ( ! [W: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ W @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_6730_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_6731_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6732_sum_OatMost__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( G @ zero_zero_nat )
        @ ( groups7108830773950497114d_enat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6733_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6734_sum_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_6735_sum__up__index__split,axiom,
    ! [F: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6736_sum__up__index__split,axiom,
    ! [F: nat > extended_enat,M: nat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups7108830773950497114d_enat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6737_sum__up__index__split,axiom,
    ! [F: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6738_sum__up__index__split,axiom,
    ! [F: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6739_prod_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6740_prod_OatMost__shift,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_complex @ ( G @ zero_zero_nat )
        @ ( groups6464643781859351333omplex
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6741_prod_OatMost__shift,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( G @ zero_zero_nat )
        @ ( groups7961826882256487087d_enat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6742_prod_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6743_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_6744_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_6745_gbinomial__parallel__sum,axiom,
    ! [A: complex,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_6746_gbinomial__parallel__sum,axiom,
    ! [A: real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_6747_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_6748_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > real,N2: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_6749_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_6750_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_6751_sum__gp__basic,axiom,
    ! [X: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6752_sum__gp__basic,axiom,
    ! [X: complex,N2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6753_sum__gp__basic,axiom,
    ! [X: real,N2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_6754_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ X2 @ I5 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_complex ) ) )
      = ( ? [I5: nat] :
            ( ( ord_less_eq_nat @ I5 @ N2 )
            & ( ( C @ I5 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_6755_polyfun__finite__roots,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ X2 @ I5 ) )
                @ ( set_ord_atMost_nat @ N2 ) )
              = zero_zero_real ) ) )
      = ( ? [I5: nat] :
            ( ( ord_less_eq_nat @ I5 @ N2 )
            & ( ( C @ I5 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_6756_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ Z6 @ I5 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_6757_polyfun__roots__finite,axiom,
    ! [C: nat > real,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z6: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ Z6 @ I5 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_6758_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N2: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( times_times_int @ ( C @ I5 ) @ ( power_power_int @ A @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int )
     => ~ ! [B4: nat > int] :
            ~ ! [Z4: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I5: nat] : ( times_times_int @ ( C @ I5 ) @ ( power_power_int @ Z4 @ I5 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_int @ ( minus_minus_int @ Z4 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I5: nat] : ( times_times_int @ ( B4 @ I5 ) @ ( power_power_int @ Z4 @ I5 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6759_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N2: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ A @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex )
     => ~ ! [B4: nat > complex] :
            ~ ! [Z4: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ Z4 @ I5 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I5: nat] : ( times_times_complex @ ( B4 @ I5 ) @ ( power_power_complex @ Z4 @ I5 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6760_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N2: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ A @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real )
     => ~ ! [B4: nat > real] :
            ~ ! [Z4: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ Z4 @ I5 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_real @ ( minus_minus_real @ Z4 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( B4 @ I5 ) @ ( power_power_real @ Z4 @ I5 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6761_polyfun__linear__factor,axiom,
    ! [C: nat > int,N2: nat,A: int] :
    ? [B4: nat > int] :
    ! [Z4: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( times_times_int @ ( C @ I5 ) @ ( power_power_int @ Z4 @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( times_times_int @ ( B4 @ I5 ) @ ( power_power_int @ Z4 @ I5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( times_times_int @ ( C @ I5 ) @ ( power_power_int @ A @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6762_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N2: nat,A: complex] :
    ? [B4: nat > complex] :
    ! [Z4: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ Z4 @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( B4 @ I5 ) @ ( power_power_complex @ Z4 @ I5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ A @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6763_polyfun__linear__factor,axiom,
    ! [C: nat > real,N2: nat,A: real] :
    ? [B4: nat > real] :
    ! [Z4: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ Z4 @ I5 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( B4 @ I5 ) @ ( power_power_real @ Z4 @ I5 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ A @ I5 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6764_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6765_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6766_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6767_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_6768_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > real,N2: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_6769_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_6770_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I5: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I5 @ J3 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I5: nat] : ( G @ I5 @ ( minus_minus_nat @ K2 @ I5 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_6771_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6772_sum_Oin__pairs__0,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7108830773950497114d_enat
        @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6773_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6774_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_6775_polynomial__product,axiom,
    ! [M: nat,A: nat > int,N2: nat,B: nat > int,X: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( power_power_int @ X @ I5 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R4: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_int @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6776_polynomial__product,axiom,
    ! [M: nat,A: nat > complex,N2: nat,B: nat > complex,X: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I5: nat] : ( times_times_complex @ ( A @ I5 ) @ ( power_power_complex @ X @ I5 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R4: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_complex @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6777_polynomial__product,axiom,
    ! [M: nat,A: nat > real,N2: nat,B: nat > real,X: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( power_power_real @ X @ I5 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R4: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_real @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6778_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I5: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6779_prod_Oin__pairs__0,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [I5: nat] : ( times_times_complex @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6780_prod_Oin__pairs__0,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups7961826882256487087d_enat
        @ ^ [I5: nat] : ( times_7803423173614009249d_enat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6781_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6782_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_6783_polyfun__eq__const,axiom,
    ! [C: nat > complex,N2: nat,K: complex] :
      ( ( ! [X2: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X2 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_6784_polyfun__eq__const,axiom,
    ! [C: nat > real,N2: nat,K: real] :
      ( ( ! [X2: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ X2 @ I5 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X2 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_6785_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I5: nat] : ( times_times_nat @ ( A @ I5 ) @ ( power_power_nat @ X @ I5 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R4: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R4 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R4 ) )
                @ ( power_power_nat @ X @ R4 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_6786_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_6787_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6788_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6789_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > extended_enat,H2: nat > extended_enat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups7108830773950497114d_enat
            @ ^ [J3: nat] : ( if_Extended_enat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_Extended_enat @ ( J3 = K ) @ zero_z5237406670263579293d_enat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups7108830773950497114d_enat
            @ ^ [J3: nat] : ( if_Extended_enat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6790_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6791_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_6792_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6793_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6794_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6795_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_6796_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_6797_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_6798_root__polyfun,axiom,
    ! [N2: nat,Z: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_complex @ Z @ N2 )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( if_complex @ ( I5 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I5 = N2 ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_6799_root__polyfun,axiom,
    ! [N2: nat,Z: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_int @ Z @ N2 )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( times_times_int @ ( if_int @ ( I5 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I5 = N2 ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_6800_root__polyfun,axiom,
    ! [N2: nat,Z: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_real @ Z @ N2 )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( if_real @ ( I5 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I5 = N2 ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_6801_sum__gp0,axiom,
    ! [X: complex,N2: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_6802_sum__gp0,axiom,
    ! [X: real,N2: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_6803_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_6804_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_6805_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6806_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6807_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( power_power_int @ X @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( power_power_int @ Y @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K2 ) ) @ ( power_power_int @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_6808_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( A @ I5 ) @ ( power_power_complex @ X @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( A @ I5 ) @ ( power_power_complex @ Y @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K2 ) ) @ ( power_power_complex @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_6809_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( power_power_real @ X @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( power_power_real @ Y @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K2 ) ) @ ( power_power_real @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_6810_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > complex,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M8: real] :
        ! [Z4: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ Z4 @ I5 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_6811_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M8: real] :
        ! [Z4: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ Z4 @ I5 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z4 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_6812_polyfun__diff,axiom,
    ! [N2: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( power_power_int @ X @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( power_power_int @ Y @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I5 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_int @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_6813_polyfun__diff,axiom,
    ! [N2: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( A @ I5 ) @ ( power_power_complex @ X @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( A @ I5 ) @ ( power_power_complex @ Y @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I5: nat] : ( times_times_complex @ ( A @ I5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I5 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_complex @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_6814_polyfun__diff,axiom,
    ! [N2: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( power_power_real @ X @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( power_power_real @ Y @ I5 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I5 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_real @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_6815_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_6816_sin__cos__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% sin_cos_npi
thf(fact_6817_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_6818_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I5 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_6819_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I5 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_6820_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I5 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_6821_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I5: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I5 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_6822_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I5 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_6823_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I5 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_6824_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_6825_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K2: nat] :
          ( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_6826_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K2: nat] :
          ( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_6827_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_6828_sin__zero,axiom,
    ( ( sin_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sin_zero
thf(fact_6829_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_6830_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_6831_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_6832_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_6833_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_6834_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_6835_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_6836_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_6837_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_6838_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_6839_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_6840_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_6841_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_6842_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_6843_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_6844_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_6845_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_6846_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri4449623510593786356d_enat @ ( suc @ N2 ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ ( suc @ N2 ) ) @ ( semiri4449623510593786356d_enat @ N2 ) ) ) ).

% fact_Suc
thf(fact_6847_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_6848_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_6849_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_6850_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq_nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_6851_sin__of__real__pi,axiom,
    ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
    = zero_zero_real ) ).

% sin_of_real_pi
thf(fact_6852_sin__of__real__pi,axiom,
    ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = zero_zero_complex ) ).

% sin_of_real_pi
thf(fact_6853_sin__cos__squared__add3,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_6854_sin__cos__squared__add3,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_6855_sin__cos__squared__add,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_6856_sin__cos__squared__add,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_6857_sin__cos__squared__add2,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_6858_sin__cos__squared__add2,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_6859_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_6860_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_6861_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
     => ( ( sin_real @ X )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_6862_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
        = one_one_complex )
     => ( ( sin_complex @ X )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_6863_sin__add,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_6864_sin__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( sin_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% sin_add
thf(fact_6865_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_6866_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_6867_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ N2 )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_6868_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ N2 )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_6869_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri4449623510593786356d_enat @ N2 )
     != zero_z5237406670263579293d_enat ) ).

% fact_nonzero
thf(fact_6870_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ N2 )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_6871_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ N2 )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_6872_cos__diff,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_6873_cos__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_diff
thf(fact_6874_cos__add,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_6875_cos__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_add
thf(fact_6876_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_6877_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( ( sin_complex @ X )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_6878_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_6879_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_6880_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6881_Suc__times__binomial__eq,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_6882_Suc__times__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_6883_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_6884_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_6885_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_6886_binomial__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_6887_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_6888_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_6889_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_6890_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_6891_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_6892_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_6893_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_6894_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_6895_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_6896_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_6897_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_6898_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_6899_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_mono
thf(fact_6900_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono
thf(fact_6901_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_mono
thf(fact_6902_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_6903_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_6904_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_6905_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_6906_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_6907_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_6908_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_6909_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_6910_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_6911_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_6912_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_6913_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_6914_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_6915_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_6916_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_6917_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_6918_cos__diff__cos,axiom,
    ! [W2: complex,Z: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_6919_cos__diff__cos,axiom,
    ! [W2: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_6920_sin__diff__sin,axiom,
    ! [W2: complex,Z: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_6921_sin__diff__sin,axiom,
    ! [W2: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_6922_sin__plus__sin,axiom,
    ! [W2: complex,Z: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_6923_sin__plus__sin,axiom,
    ! [W2: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_6924_cos__times__sin,axiom,
    ! [W2: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_6925_cos__times__sin,axiom,
    ! [W2: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_6926_sin__times__cos,axiom,
    ! [W2: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_6927_sin__times__cos,axiom,
    ! [W2: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_6928_sin__times__sin,axiom,
    ! [W2: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_6929_sin__times__sin,axiom,
    ! [W2: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_6930_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6931_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6932_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_6933_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_6934_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_6935_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_6936_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_6937_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_6938_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ K2 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_6939_minus__sin__cos__eq,axiom,
    ! [X: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_6940_minus__sin__cos__eq,axiom,
    ! [X: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
      = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_6941_binomial__absorption,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_6942_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_6943_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R2 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_6944_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% choose_dvd
thf(fact_6945_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% choose_dvd
thf(fact_6946_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% choose_dvd
thf(fact_6947_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_6948_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_6949_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_6950_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_6951_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_6952_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K2 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).

% binomial_code
thf(fact_6953_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_6954_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% binomial_maximum'
thf(fact_6955_binomial__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_6956_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_6957_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_6958_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_6959_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_6960_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_6961_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_6962_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_6963_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_6964_vandermonde,axiom,
    ! [M: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_6965_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_6966_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_6967_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N2 )
         => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_6968_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_6969_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_6970_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_6971_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_6972_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6973_fact__num__eq__if,axiom,
    ( semiri4449623510593786356d_enat
    = ( ^ [M2: nat] : ( if_Extended_enat @ ( M2 = zero_zero_nat ) @ one_on7984719198319812577d_enat @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ M2 ) @ ( semiri4449623510593786356d_enat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6974_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6975_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6976_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6977_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri5044797733671781792omplex @ N2 )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6978_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri4449623510593786356d_enat @ N2 )
        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( semiri4449623510593786356d_enat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6979_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6980_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6981_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri2265585572941072030t_real @ N2 )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_6982_cos__plus__cos,axiom,
    ! [W2: complex,Z: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_6983_cos__plus__cos,axiom,
    ! [W2: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_6984_cos__times__cos,axiom,
    ! [W2: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_6985_cos__times__cos,axiom,
    ! [W2: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_6986_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_6987_binomial__ring,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6988_binomial__ring,axiom,
    ! [A: extended_enat,B: extended_enat,N2: nat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ N2 )
      = ( groups7108830773950497114d_enat
        @ ^ [K2: nat] : ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ ( binomial @ N2 @ K2 ) ) @ ( power_8040749407984259932d_enat @ A @ K2 ) ) @ ( power_8040749407984259932d_enat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6989_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6990_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6991_binomial__ring,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_6992_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K2 ) ) @ ( comm_s2602460028002588243omplex @ A @ K2 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_6993_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_6994_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_6995_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_6996_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_6997_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_6998_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_6999_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_7000_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > complex > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_7001_Maclaurin__zero,axiom,
    ! [X: real,N2: nat,Diff: nat > extended_enat > real] :
      ( ( X = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_z5237406670263579293d_enat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_z5237406670263579293d_enat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7002_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_7003_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_7004_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_7005_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I5 ) @ ( semiri8010041392384452111omplex @ I5 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I5 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_7006_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I5 ) @ ( semiri1314217659103216013at_int @ I5 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I5 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_7007_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( semiri5074537144036343181t_real @ I5 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I5 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_7008_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7009_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I5: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7010_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I5: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_7011_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_7012_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I5 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I5 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_7013_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I5 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I5 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_7014_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I5 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_7015_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_7016_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P6: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_complex @ X @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_7017_sin__x__sin__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P6: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) ) @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_7018_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P6: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_7019_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P6: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_7020_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P6: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_complex @ X @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_7021_cos__x__cos__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P6: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P6 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P6 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P6 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P6 ) )
      @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_7022_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).

% sin_coeff_def
thf(fact_7023_tan__double,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7024_tan__double,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7025_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_7026_scaleR__zero__right,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% scaleR_zero_right
thf(fact_7027_scaleR__zero__right,axiom,
    ! [A: real] :
      ( ( real_V2046097035970521341omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% scaleR_zero_right
thf(fact_7028_scaleR__cancel__right,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ( real_V1485227260804924795R_real @ A @ X )
        = ( real_V1485227260804924795R_real @ B @ X ) )
      = ( ( A = B )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_cancel_right
thf(fact_7029_scaleR__cancel__right,axiom,
    ! [A: real,X: complex,B: real] :
      ( ( ( real_V2046097035970521341omplex @ A @ X )
        = ( real_V2046097035970521341omplex @ B @ X ) )
      = ( ( A = B )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_cancel_right
thf(fact_7030_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_7031_tan__zero,axiom,
    ( ( tan_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tan_zero
thf(fact_7032_scaleR__zero__left,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ zero_zero_real @ X )
      = zero_zero_real ) ).

% scaleR_zero_left
thf(fact_7033_scaleR__zero__left,axiom,
    ! [X: complex] :
      ( ( real_V2046097035970521341omplex @ zero_zero_real @ X )
      = zero_zero_complex ) ).

% scaleR_zero_left
thf(fact_7034_scaleR__eq__0__iff,axiom,
    ! [A: real,X: real] :
      ( ( ( real_V1485227260804924795R_real @ A @ X )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( X = zero_zero_real ) ) ) ).

% scaleR_eq_0_iff
thf(fact_7035_scaleR__eq__0__iff,axiom,
    ! [A: real,X: complex] :
      ( ( ( real_V2046097035970521341omplex @ A @ X )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        | ( X = zero_zero_complex ) ) ) ).

% scaleR_eq_0_iff
thf(fact_7036_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
        = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
      = ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_7037_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_7038_scaleR__collapse,axiom,
    ! [U: real,A: real] :
      ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_7039_scaleR__half__double,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_7040_scaleR__right__imp__eq,axiom,
    ! [X: real,A: real,B: real] :
      ( ( X != zero_zero_real )
     => ( ( ( real_V1485227260804924795R_real @ A @ X )
          = ( real_V1485227260804924795R_real @ B @ X ) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
thf(fact_7041_scaleR__right__imp__eq,axiom,
    ! [X: complex,A: real,B: real] :
      ( ( X != zero_zero_complex )
     => ( ( ( real_V2046097035970521341omplex @ A @ X )
          = ( real_V2046097035970521341omplex @ B @ X ) )
       => ( A = B ) ) ) ).

% scaleR_right_imp_eq
thf(fact_7042_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).

% scaleR_right_distrib
thf(fact_7043_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).

% scaleR_left_distrib
thf(fact_7044_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y ) @ Xa2 )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y @ Xa2 ) ) ) ).

% scaleR_left.add
thf(fact_7045_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).

% scaleR_right_mono_neg
thf(fact_7046_scaleR__right__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).

% scaleR_right_mono
thf(fact_7047_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% scaleR_le_cancel_left_pos
thf(fact_7048_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% scaleR_le_cancel_left_neg
thf(fact_7049_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% scaleR_le_cancel_left
thf(fact_7050_scaleR__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).

% scaleR_left_mono_neg
thf(fact_7051_scaleR__left__mono,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ) ).

% scaleR_left_mono
thf(fact_7052_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: real,X: real] :
      ( ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A )
        = X )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ U @ A )
            = ( real_V1485227260804924795R_real @ V @ X ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_7053_vector__fraction__eq__iff,axiom,
    ! [U: real,V: real,A: complex,X: complex] :
      ( ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A )
        = X )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V2046097035970521341omplex @ U @ A )
            = ( real_V2046097035970521341omplex @ V @ X ) ) ) ) ) ).

% vector_fraction_eq_iff
thf(fact_7054_eq__vector__fraction__iff,axiom,
    ! [X: real,U: real,V: real,A: real] :
      ( ( X
        = ( real_V1485227260804924795R_real @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_real ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V1485227260804924795R_real @ V @ X )
            = ( real_V1485227260804924795R_real @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_7055_eq__vector__fraction__iff,axiom,
    ! [X: complex,U: real,V: real,A: complex] :
      ( ( X
        = ( real_V2046097035970521341omplex @ ( divide_divide_real @ U @ V ) @ A ) )
      = ( ( ( V = zero_zero_real )
         => ( X = zero_zero_complex ) )
        & ( ( V != zero_zero_real )
         => ( ( real_V2046097035970521341omplex @ V @ X )
            = ( real_V2046097035970521341omplex @ U @ A ) ) ) ) ) ).

% eq_vector_fraction_iff
thf(fact_7056_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_7057_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_7058_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
thf(fact_7059_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
thf(fact_7060_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).

% scaleR_nonpos_nonpos
thf(fact_7061_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).

% scaleR_nonpos_nonneg
thf(fact_7062_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).

% scaleR_nonneg_nonpos
thf(fact_7063_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).

% scaleR_nonneg_nonneg
thf(fact_7064_split__scaleR__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).

% split_scaleR_pos_le
thf(fact_7065_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ X @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ X ) ) )
     => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).

% split_scaleR_neg_le
thf(fact_7066_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).

% scaleR_mono'
thf(fact_7067_scaleR__mono,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ X )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).

% scaleR_mono
thf(fact_7068_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).

% scaleR_left_le_one_le
thf(fact_7069_scaleR__2,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
      = ( plus_plus_real @ X @ X ) ) ).

% scaleR_2
thf(fact_7070_sin__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( sin_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_7071_add__tan__eq,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7072_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7073_lemma__tan__add1,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7074_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7075_tan__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7076_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7077_tan__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7078_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7079_cos__coeff__Suc,axiom,
    ! [N2: nat] :
      ( ( cos_coeff @ ( suc @ N2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_7080_tan__half,axiom,
    ( tan_complex
    = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_7081_tan__half,axiom,
    ( tan_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_7082_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_7083_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = 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 @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = 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 @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( 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 @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_7084_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = 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 @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = 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 @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( 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 @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_7085_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_7086_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_7087_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_7088_sgn__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% sgn_sgn
thf(fact_7089_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_7090_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_7091_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_7092_sgn__zero,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_zero
thf(fact_7093_sgn__zero,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_zero
thf(fact_7094_sgn__1,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_1
thf(fact_7095_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_7096_sgn__1,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_1
thf(fact_7097_bit_Oxor__self,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ X )
      = zero_zero_int ) ).

% bit.xor_self
thf(fact_7098_xor__self__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ A )
      = zero_zero_int ) ).

% xor_self_eq
thf(fact_7099_xor__self__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ A )
      = zero_zero_nat ) ).

% xor_self_eq
thf(fact_7100_xor_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
      = A ) ).

% xor.left_neutral
thf(fact_7101_xor_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
      = A ) ).

% xor.left_neutral
thf(fact_7102_xor_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
      = A ) ).

% xor.right_neutral
thf(fact_7103_xor_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
      = A ) ).

% xor.right_neutral
thf(fact_7104_sgn__divide,axiom,
    ! [A: real,B: real] :
      ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).

% sgn_divide
thf(fact_7105_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_7106_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_7107_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R4: nat] : ( if_nat @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7108_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R4: nat] : ( if_real @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7109_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R4: nat] : ( if_int @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7110_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R4: nat] : ( if_complex @ ( R4 = I ) @ ( F @ R4 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_7111_summable__zero,axiom,
    ( summable_nat
    @ ^ [N: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7112_summable__zero,axiom,
    ( summable_real
    @ ^ [N: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7113_summable__zero,axiom,
    ( summable_int
    @ ^ [N: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7114_summable__zero,axiom,
    ( summable_complex
    @ ^ [N: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_7115_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7116_sgn__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_less
thf(fact_7117_sgn__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_less
thf(fact_7118_sgn__greater,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_greater
thf(fact_7119_sgn__greater,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_greater
thf(fact_7120_divide__sgn,axiom,
    ! [A: real,B: real] :
      ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
      = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).

% divide_sgn
thf(fact_7121_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7122_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7123_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7124_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7125_summable__If__finite,axiom,
    ! [P2: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( summable_nat
        @ ^ [R4: nat] : ( if_nat @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_7126_summable__If__finite,axiom,
    ! [P2: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( summable_real
        @ ^ [R4: nat] : ( if_real @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_7127_summable__If__finite,axiom,
    ! [P2: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( summable_int
        @ ^ [R4: nat] : ( if_int @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_7128_summable__If__finite,axiom,
    ! [P2: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( summable_complex
        @ ^ [R4: nat] : ( if_complex @ ( P2 @ R4 ) @ ( F @ R4 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_7129_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_nat
        @ ^ [R4: nat] : ( if_nat @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_7130_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_real
        @ ^ [R4: nat] : ( if_real @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_7131_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_int
        @ ^ [R4: nat] : ( if_int @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_7132_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_complex
        @ ^ [R4: nat] : ( if_complex @ ( member_nat2 @ R4 @ A2 ) @ ( F @ R4 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_7133_sgn__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( sgn_sgn_real @ A )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_7134_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_7135_sgn__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_mult_self_eq
thf(fact_7136_sgn__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_7137_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_7138_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_7139_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_7140_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_7141_sgn__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_7142_sgn__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% sgn_neg
thf(fact_7143_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_7144_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_7145_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_7146_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_7147_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_7148_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_7149_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_7150_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_7151_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_7152_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_7153_sgn__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sgn_sgn_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sgn_zero_iff
thf(fact_7154_sgn__zero__iff,axiom,
    ! [X: complex] :
      ( ( ( sgn_sgn_complex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_complex ) ) ).

% sgn_zero_iff
thf(fact_7155_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_7156_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_7157_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( sgn_sgn_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_7158_sgn__0__0,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_7159_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_7160_sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).

% sgn_mult
thf(fact_7161_sgn__mult,axiom,
    ! [A: real,B: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).

% sgn_mult
thf(fact_7162_sgn__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).

% sgn_mult
thf(fact_7163_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_7164_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
        = ( sgn_sgn_real @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_7165_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7166_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_7167_summable__comparison__test_H,axiom,
    ! [G: nat > real,N7: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7168_summable__comparison__test_H,axiom,
    ! [G: nat > real,N7: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7169_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7170_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7171_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7172_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7173_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7174_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7175_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7176_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7177_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7178_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7179_summable__finite,axiom,
    ! [N7: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_7180_summable__finite,axiom,
    ! [N7: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_7181_summable__finite,axiom,
    ! [N7: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_7182_summable__finite,axiom,
    ! [N7: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat2 @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( summable_complex @ F ) ) ) ).

% summable_finite
thf(fact_7183_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
       != ( sgn_sgn_int @ A ) )
     => ( ( ( sgn_sgn_int @ A )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_7184_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
       != ( sgn_sgn_real @ A ) )
     => ( ( ( sgn_sgn_real @ A )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_7185_sgn__minus__1,axiom,
    ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% sgn_minus_1
thf(fact_7186_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_7187_sgn__minus__1,axiom,
    ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sgn_minus_1
thf(fact_7188_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N3: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_sgnE
thf(fact_7189_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7190_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7191_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7192_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7193_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7194_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7195_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7196_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7197_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_7198_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_7199_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_7200_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 )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7201_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 )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7202_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 )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7203_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_7204_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_7205_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_7206_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = one_one_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_1_pos
thf(fact_7207_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_1_pos
thf(fact_7208_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).

% summable_zero_power'
thf(fact_7209_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_zero_power'
thf(fact_7210_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_zero_power'
thf(fact_7211_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_0_powser
thf(fact_7212_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_0_powser
thf(fact_7213_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7214_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7215_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7216_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7217_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7218_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7219_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7220_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_7221_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_7222_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_7223_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 ) )
          = ( ? [I5: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I5 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7224_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 ) )
          = ( ? [I5: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I5 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7225_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 ) )
          = ( ? [I5: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I5 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7226_suminf__le__const,axiom,
    ! [F: nat > int,X: int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7227_suminf__le__const,axiom,
    ! [F: nat > nat,X: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7228_suminf__le__const,axiom,
    ! [F: nat > real,X: real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_7229_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_7230_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_1_neg
thf(fact_7231_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X2: int] : ( if_int @ ( X2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_7232_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X2: real] : ( if_real @ ( X2 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X2 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_7233_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I5: int] : ( if_int @ ( I5 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I5 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_7234_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X: 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 ) ) @ X )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7235_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X: 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 ) ) @ X )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7236_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X: 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 ) ) @ X )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7237_norm__sgn,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_7238_norm__sgn,axiom,
    ! [X: complex] :
      ( ( ( X = zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
          = zero_zero_real ) )
      & ( ( X != zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_7239_bounded__imp__summable,axiom,
    ! [A: nat > int,B2: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7240_bounded__imp__summable,axiom,
    ! [A: nat > nat,B2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7241_bounded__imp__summable,axiom,
    ! [A: nat > real,B2: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B2 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7242_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7243_sum__le__suminf,axiom,
    ! [F: nat > int,I6: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N3: nat] :
              ( ( member_nat2 @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7244_sum__le__suminf,axiom,
    ! [F: nat > nat,I6: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N3: nat] :
              ( ( member_nat2 @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7245_sum__le__suminf,axiom,
    ! [F: nat > real,I6: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N3: nat] :
              ( ( member_nat2 @ N3 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7246_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7247_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7248_sum__less__suminf,axiom,
    ! [F: nat > int,N2: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7249_sum__less__suminf,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7250_sum__less__suminf,axiom,
    ! [F: nat > real,N2: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7251_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7252_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7253_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7254_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7255_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E2: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M4: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M4 )
                 => ! [N6: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M4 @ N6 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7256_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E2: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M4: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M4 )
                 => ! [N6: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M4 @ N6 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7257_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N9 @ N6 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N6 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7258_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N9 @ N6 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I5: nat] : ( F @ ( plus_plus_nat @ I5 @ N6 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7259_summable__ratio__test,axiom,
    ! [C: real,N7: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ 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_7260_summable__ratio__test,axiom,
    ! [C: real,N7: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ 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_7261_sum__less__suminf2,axiom,
    ! [F: nat > int,N2: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7262_sum__less__suminf2,axiom,
    ! [F: nat > nat,N2: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7263_sum__less__suminf2,axiom,
    ! [F: nat > real,N2: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N2 @ M3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7264_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = 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 @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_7265_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_7266_one__xor__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7267_one__xor__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_7268_xor__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7269_xor__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_7270_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7271_diffs__equiv,axiom,
    ! [C: nat > complex,X: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_7272_diffs__equiv,axiom,
    ! [C: nat > real,X: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_7273_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_7274_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_7275_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_7276_case__prod__Pair__iden,axiom,
    ! [P4: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_7277_case__prod__Pair__iden,axiom,
    ! [P4: produc859450856879609959at_nat] :
      ( ( produc6744312701629110395at_nat @ produc6161850002892822231at_nat @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_7278_case__prod__Pair__iden,axiom,
    ! [P4: produc9072475918466114483BT_nat] :
      ( ( produc2645369811736392845BT_nat @ produc738532404422230701BT_nat @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_7279_case__prod__Pair__iden,axiom,
    ! [P4: product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_7280_case__prod__Pair__iden,axiom,
    ! [P4: produc7272778201969148633d_enat] :
      ( ( produc4174022389229927035d_enat @ produc581526299967858633d_enat @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_7281_the__elem__eq,axiom,
    ! [X: real] :
      ( ( the_elem_real @ ( insert_real2 @ X @ bot_bot_set_real ) )
      = X ) ).

% the_elem_eq
thf(fact_7282_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o2 @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_7283_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_7284_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int2 @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_7285_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_7286_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_7287_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_7288_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_7289_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_7290_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_7291_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_7292_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_7293_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_7294_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_7295_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_7296_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_7297_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_7298_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_7299_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_7300_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_7301_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_7302_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_7303_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_7304_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_7305_abs__divide,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_divide
thf(fact_7306_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_7307_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_7308_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_7309_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_7310_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_7311_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_7312_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_7313_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_7314_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_7315_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_7316_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).

% of_int_abs
thf(fact_7317_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
      = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).

% of_int_abs
thf(fact_7318_abs__bool__eq,axiom,
    ! [P2: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( zero_n3304061248610475627l_real @ P2 ) ) ).

% abs_bool_eq
thf(fact_7319_abs__bool__eq,axiom,
    ! [P2: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% abs_bool_eq
thf(fact_7320_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_7321_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_7322_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_7323_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_7324_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7325_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7326_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_7327_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_7328_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( abs_abs_real @ ( F @ I5 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_7329_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_7330_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_7331_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_7332_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_7333_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
        = one_one_real ) ) ).

% abs_sgn_eq_1
thf(fact_7334_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_7335_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_7336_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_7337_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_7338_sgn__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_abs
thf(fact_7339_sgn__abs,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% sgn_abs
thf(fact_7340_sgn__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_abs
thf(fact_7341_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( abs_abs_real @ ( F @ I5 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_7342_zero__less__power__abs__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
      = ( ( A != zero_zero_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7343_zero__less__power__abs__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
      = ( ( A != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7344_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_7345_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_7346_norm__of__real__addn,axiom,
    ! [X: real,B: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_7347_norm__of__real__addn,axiom,
    ! [X: real,B: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_7348_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_7349_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_7350_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_7351_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_7352_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_7353_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_7354_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_7355_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_7356_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_7357_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_7358_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_7359_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_7360_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_7361_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_7362_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_7363_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_7364_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_7365_abs__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_mult
thf(fact_7366_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_7367_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_7368_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_7369_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_7370_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_7371_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7372_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7373_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_7374_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_7375_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7376_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7377_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_7378_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_7379_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_7380_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_7381_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7382_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7383_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_7384_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_7385_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_7386_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_7387_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_7388_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_7389_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_7390_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_7391_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_7392_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_7393_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_7394_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K2: int] : ( times_times_int @ K2 @ ( sgn_sgn_int @ K2 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_7395_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_real
    = ( ^ [K2: real] : ( times_times_real @ K2 @ ( sgn_sgn_real @ K2 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_7396_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_7397_abs__mult__sgn,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_7398_abs__mult__sgn,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_7399_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_7400_sgn__mult__abs,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_7401_sgn__mult__abs,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_7402_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_7403_mult__sgn__abs,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ X ) @ ( abs_abs_real @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_7404_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_7405_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_7406_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_7407_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_7408_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_7409_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_7410_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_7411_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_7412_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_7413_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7414_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7415_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7416_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7417_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_7418_zero__le__power__abs,axiom,
    ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7419_zero__le__power__abs,axiom,
    ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7420_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_7421_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_7422_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_7423_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_7424_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_7425_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_7426_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_7427_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_7428_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7429_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7430_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7431_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7432_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7433_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7434_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = one_one_real ) ) ) ).

% abs_sgn_eq
thf(fact_7435_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_7436_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7437_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7438_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7439_of__int__leD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_7440_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_7441_of__int__lessD,axiom,
    ! [N2: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_7442_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_7443_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_7444_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C2: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7445_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C2: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7446_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C2: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_7447_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7448_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7449_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7450_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7451_abs__sqrt__wlog,axiom,
    ! [P2: real > real > $o,X: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P2 @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P2 @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7452_abs__sqrt__wlog,axiom,
    ! [P2: int > int > $o,X: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P2 @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P2 @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7453_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_7454_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_7455_abs__square__less__1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_7456_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_7457_power__mono__even,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7458_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7459_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ I6 )
          = one_one_real )
       => ( ! [I3: real] :
              ( ( member_real2 @ I3 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I5: real] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7460_convex__sum__bound__le,axiom,
    ! [I6: set_o,X: $o > real,A: $o > real,B: real,Delta: real] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups8691415230153176458o_real @ X @ I6 )
          = one_one_real )
       => ( ! [I3: $o] :
              ( ( member_o2 @ I3 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8691415230153176458o_real
                  @ ^ [I5: $o] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7461_convex__sum__bound__le,axiom,
    ! [I6: set_set_nat,X: set_nat > real,A: set_nat > real,B: real,Delta: real] :
      ( ! [I3: set_nat] :
          ( ( member_set_nat2 @ I3 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups5107569545109728110t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I3: set_nat] :
              ( ( member_set_nat2 @ I3 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5107569545109728110t_real
                  @ ^ [I5: set_nat] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7462_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I3: int] :
              ( ( member_int2 @ I3 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I5: int] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7463_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > int,A: real > int,B: int,Delta: int] :
      ( ! [I3: real] :
          ( ( member_real2 @ I3 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups1932886352136224148al_int @ X @ I6 )
          = one_one_int )
       => ( ! [I3: real] :
              ( ( member_real2 @ I3 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups1932886352136224148al_int
                  @ ^ [I5: real] : ( times_times_int @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7464_convex__sum__bound__le,axiom,
    ! [I6: set_o,X: $o > int,A: $o > int,B: int,Delta: int] :
      ( ! [I3: $o] :
          ( ( member_o2 @ I3 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups8505340233167759370_o_int @ X @ I6 )
          = one_one_int )
       => ( ! [I3: $o] :
              ( ( member_o2 @ I3 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups8505340233167759370_o_int
                  @ ^ [I5: $o] : ( times_times_int @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7465_convex__sum__bound__le,axiom,
    ! [I6: set_set_nat,X: set_nat > int,A: set_nat > int,B: int,Delta: int] :
      ( ! [I3: set_nat] :
          ( ( member_set_nat2 @ I3 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups8292507037921071086at_int @ X @ I6 )
          = one_one_int )
       => ( ! [I3: set_nat] :
              ( ( member_set_nat2 @ I3 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups8292507037921071086at_int
                  @ ^ [I5: set_nat] : ( times_times_int @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7466_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > int,A: nat > int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( ( member_nat2 @ I3 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups3539618377306564664at_int @ X @ I6 )
          = one_one_int )
       => ( ! [I3: nat] :
              ( ( member_nat2 @ I3 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I5: nat] : ( times_times_int @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7467_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > int,A: int > int,B: int,Delta: int] :
      ( ! [I3: int] :
          ( ( member_int2 @ I3 @ I6 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I3 ) ) )
     => ( ( ( groups4538972089207619220nt_int @ X @ I6 )
          = one_one_int )
       => ( ! [I3: int] :
              ( ( member_int2 @ I3 @ I6 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups4538972089207619220nt_int
                  @ ^ [I5: int] : ( times_times_int @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7468_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > real,A: nat > real,B: real,Delta: real] :
      ( ! [I3: nat] :
          ( ( member_nat2 @ I3 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups6591440286371151544t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I3: nat] :
              ( ( member_nat2 @ I3 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7469_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_7470_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_7471_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_7472_of__int__round__abs__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7473_round__unique_H,axiom,
    ! [X: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_7474_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7475_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7476_Maclaurin__exp__lt,axiom,
    ! [X: real,N2: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
            & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_7477_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N2: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_7478_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_7479_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_7480_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_7481_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_7482_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_7483_exp__not__eq__zero,axiom,
    ! [X: real] :
      ( ( exp_real @ X )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_7484_exp__not__eq__zero,axiom,
    ! [X: complex] :
      ( ( exp_complex @ X )
     != zero_zero_complex ) ).

% exp_not_eq_zero
thf(fact_7485_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( abs_abs_int @ A )
          = ( abs_abs_int @ B ) ) ) ) ).

% zdvd_antisym_abs
thf(fact_7486_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
        = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7487_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
        = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7488_mult__exp__exp,axiom,
    ! [X: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7489_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
      = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7490_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7491_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I5: int] : ( if_int @ ( ord_less_int @ I5 @ zero_zero_int ) @ ( uminus_uminus_int @ I5 ) @ I5 ) ) ) ).

% zabs_def
thf(fact_7492_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_7493_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
        = ( ( abs_abs_int @ N2 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7494_powr__def,axiom,
    ( powr_real
    = ( ^ [X2: real,A3: real] : ( if_real @ ( X2 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X2 ) ) ) ) ) ) ).

% powr_def
thf(fact_7495_exp__divide__power__eq,axiom,
    ! [N2: nat,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
        = ( exp_complex @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7496_exp__divide__power__eq,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
        = ( exp_real @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7497_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N2 )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7498_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7499_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7500_monoI1,axiom,
    ! [X7: nat > filter_nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_le2510731241096832064er_nat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo6427056007704750605er_nat @ X7 ) ) ).

% monoI1
thf(fact_7501_monoI1,axiom,
    ! [X7: nat > real] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% monoI1
thf(fact_7502_monoI1,axiom,
    ! [X7: nat > set_nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_set_nat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo7278393974255667507et_nat @ X7 ) ) ).

% monoI1
thf(fact_7503_monoI1,axiom,
    ! [X7: nat > nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% monoI1
thf(fact_7504_monoI1,axiom,
    ! [X7: nat > int] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% monoI1
thf(fact_7505_monoI2,axiom,
    ! [X7: nat > filter_nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_le2510731241096832064er_nat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo6427056007704750605er_nat @ X7 ) ) ).

% monoI2
thf(fact_7506_monoI2,axiom,
    ! [X7: nat > real] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% monoI2
thf(fact_7507_monoI2,axiom,
    ! [X7: nat > set_nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_set_nat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo7278393974255667507et_nat @ X7 ) ) ).

% monoI2
thf(fact_7508_monoI2,axiom,
    ! [X7: nat > nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% monoI2
thf(fact_7509_monoI2,axiom,
    ! [X7: nat > int] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% monoI2
thf(fact_7510_monoseq__def,axiom,
    ( topolo6427056007704750605er_nat
    = ( ^ [X8: nat > filter_nat] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_le2510731241096832064er_nat @ ( X8 @ M2 ) @ ( X8 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_le2510731241096832064er_nat @ ( X8 @ N ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7511_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X8: nat > real] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_real @ ( X8 @ M2 ) @ ( X8 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_real @ ( X8 @ N ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7512_monoseq__def,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X8: nat > set_nat] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_set_nat @ ( X8 @ M2 ) @ ( X8 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_set_nat @ ( X8 @ N ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7513_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X8: nat > nat] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_nat @ ( X8 @ M2 ) @ ( X8 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_nat @ ( X8 @ N ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7514_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X8: nat > int] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_int @ ( X8 @ M2 ) @ ( X8 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_int @ ( X8 @ N ) @ ( X8 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7515_monoseq__Suc,axiom,
    ( topolo6427056007704750605er_nat
    = ( ^ [X8: nat > filter_nat] :
          ( ! [N: nat] : ( ord_le2510731241096832064er_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_le2510731241096832064er_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7516_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X8: nat > real] :
          ( ! [N: nat] : ( ord_less_eq_real @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7517_monoseq__Suc,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X8: nat > set_nat] :
          ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7518_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X8: nat > nat] :
          ( ! [N: nat] : ( ord_less_eq_nat @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7519_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X8: nat > int] :
          ( ! [N: nat] : ( ord_less_eq_int @ ( X8 @ N ) @ ( X8 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N ) ) @ ( X8 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7520_mono__SucI2,axiom,
    ! [X7: nat > filter_nat] :
      ( ! [N3: nat] : ( ord_le2510731241096832064er_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo6427056007704750605er_nat @ X7 ) ) ).

% mono_SucI2
thf(fact_7521_mono__SucI2,axiom,
    ! [X7: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% mono_SucI2
thf(fact_7522_mono__SucI2,axiom,
    ! [X7: nat > set_nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo7278393974255667507et_nat @ X7 ) ) ).

% mono_SucI2
thf(fact_7523_mono__SucI2,axiom,
    ! [X7: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% mono_SucI2
thf(fact_7524_mono__SucI2,axiom,
    ! [X7: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% mono_SucI2
thf(fact_7525_mono__SucI1,axiom,
    ! [X7: nat > filter_nat] :
      ( ! [N3: nat] : ( ord_le2510731241096832064er_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo6427056007704750605er_nat @ X7 ) ) ).

% mono_SucI1
thf(fact_7526_mono__SucI1,axiom,
    ! [X7: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% mono_SucI1
thf(fact_7527_mono__SucI1,axiom,
    ! [X7: nat > set_nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo7278393974255667507et_nat @ X7 ) ) ).

% mono_SucI1
thf(fact_7528_mono__SucI1,axiom,
    ! [X7: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% mono_SucI1
thf(fact_7529_mono__SucI1,axiom,
    ! [X7: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% mono_SucI1
thf(fact_7530_exp__first__two__terms,axiom,
    ( exp_real
    = ( ^ [X2: real] :
          ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 )
          @ ( suminf_real
            @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_7531_exp__first__two__terms,axiom,
    ( exp_complex
    = ( ^ [X2: complex] :
          ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X2 )
          @ ( suminf_complex
            @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X2 @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_7532_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K2: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_7533_exp__first__terms,axiom,
    ! [K: nat] :
      ( exp_complex
      = ( ^ [X2: complex] :
            ( plus_plus_complex
            @ ( groups2073611262835488442omplex
              @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X2 @ N ) )
              @ ( set_ord_lessThan_nat @ K ) )
            @ ( suminf_complex
              @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ K ) ) ) @ ( power_power_complex @ X2 @ ( plus_plus_nat @ N @ K ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_7534_exp__first__terms,axiom,
    ! [K: nat] :
      ( exp_real
      = ( ^ [X2: real] :
            ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) )
              @ ( set_ord_lessThan_nat @ K ) )
            @ ( suminf_real
              @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ K ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ N @ K ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_7535_cosh__zero__iff,axiom,
    ! [X: complex] :
      ( ( ( cosh_complex @ X )
        = zero_zero_complex )
      = ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% cosh_zero_iff
thf(fact_7536_cosh__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cosh_real @ X )
        = zero_zero_real )
      = ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% cosh_zero_iff
thf(fact_7537_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7538_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_7539_inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_7540_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
      = ( A = B ) ) ).

% inverse_eq_iff_eq
thf(fact_7541_inverse__zero,axiom,
    ( ( invers8013647133539491842omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% inverse_zero
thf(fact_7542_inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% inverse_zero
thf(fact_7543_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7544_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7545_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_7546_inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_7547_inverse__1,axiom,
    ( ( invers8013647133539491842omplex @ one_one_complex )
    = one_one_complex ) ).

% inverse_1
thf(fact_7548_inverse__1,axiom,
    ( ( inverse_inverse_real @ one_one_real )
    = one_one_real ) ).

% inverse_1
thf(fact_7549_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( invers8013647133539491842omplex @ X )
        = one_one_complex )
      = ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
thf(fact_7550_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( inverse_inverse_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% inverse_eq_1_iff
thf(fact_7551_inverse__divide,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ B @ A ) ) ).

% inverse_divide
thf(fact_7552_inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7553_abs__inverse,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).

% abs_inverse
thf(fact_7554_inverse__sgn,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% inverse_sgn
thf(fact_7555_sgn__inverse,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).

% sgn_inverse
thf(fact_7556_nat__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
      = N2 ) ).

% nat_int
thf(fact_7557_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_7558_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_7559_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_7560_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_7561_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% inverse_negative_iff_negative
thf(fact_7562_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_7563_cosh__0,axiom,
    ( ( cosh_real @ zero_zero_real )
    = one_one_real ) ).

% cosh_0
thf(fact_7564_cosh__0,axiom,
    ( ( cosh_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cosh_0
thf(fact_7565_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_7566_nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% nat_of_bool
thf(fact_7567_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_7568_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_7569_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
        = one_one_complex ) ) ).

% right_inverse
thf(fact_7570_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_7571_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% left_inverse
thf(fact_7572_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_7573_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_7574_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_7575_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_7576_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_7577_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_7578_nat__zminus__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_7579_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_7580_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_7581_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
        = ( ring_1_of_int_real @ Z ) ) ) ).

% of_nat_nat
thf(fact_7582_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_7583_diff__nat__numeral,axiom,
    ! [V: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_7584_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_7585_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N2: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_7586_dvd__nat__abs__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( dvd_dvd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).

% dvd_nat_abs_iff
thf(fact_7587_nat__abs__dvd__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
      = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_abs_dvd_iff
thf(fact_7588_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_7589_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_7590_nat__numeral__diff__1,axiom,
    ! [V: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_7591_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_7592_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_7593_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_7594_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_7595_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,X: complex] :
      ( ( ( times_times_complex @ Y @ X )
        = ( times_times_complex @ X @ Y ) )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y ) @ X )
        = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7596_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X: real] :
      ( ( ( times_times_real @ Y @ X )
        = ( times_times_real @ X @ Y ) )
     => ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X )
        = ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7597_nonzero__norm__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( real_V7735802525324610683m_real @ A ) ) ) ) ).

% nonzero_norm_inverse
thf(fact_7598_nonzero__norm__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ A ) )
        = ( inverse_inverse_real @ ( real_V1022390504157884413omplex @ A ) ) ) ) ).

% nonzero_norm_inverse
thf(fact_7599_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
thf(fact_7600_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
       != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_7601_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
       != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_7602_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_7603_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_7604_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B ) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_7605_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_7606_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
thf(fact_7607_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
thf(fact_7608_field__class_Ofield__inverse__zero,axiom,
    ( ( invers8013647133539491842omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% field_class.field_inverse_zero
thf(fact_7609_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% field_class.field_inverse_zero
thf(fact_7610_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: complex] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_complex )
       => ( ( invers8013647133539491842omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
          = ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ A ) @ ( invers8013647133539491842omplex @ X ) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
thf(fact_7611_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( X != zero_zero_real )
       => ( ( inverse_inverse_real @ ( real_V1485227260804924795R_real @ A @ X ) )
          = ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ X ) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
thf(fact_7612_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_7613_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_7614_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_7615_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_7616_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% inverse_negative_imp_negative
thf(fact_7617_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_7618_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).

% negative_imp_inverse_negative
thf(fact_7619_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_7620_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
          = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_7621_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
          = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_7622_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
        = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_7623_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
        = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_7624_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = one_one_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = B ) ) ).

% inverse_unique
thf(fact_7625_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = one_one_real )
     => ( ( inverse_inverse_real @ A )
        = B ) ) ).

% inverse_unique
thf(fact_7626_divide__inverse__commute,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A3 ) ) ) ).

% divide_inverse_commute
thf(fact_7627_divide__inverse__commute,axiom,
    ( divide_divide_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A3 ) ) ) ).

% divide_inverse_commute
thf(fact_7628_divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).

% divide_inverse
thf(fact_7629_divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).

% divide_inverse
thf(fact_7630_field__class_Ofield__divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_7631_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B3 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_7632_inverse__eq__divide,axiom,
    ( invers8013647133539491842omplex
    = ( divide1717551699836669952omplex @ one_one_complex ) ) ).

% inverse_eq_divide
thf(fact_7633_inverse__eq__divide,axiom,
    ( inverse_inverse_real
    = ( divide_divide_real @ one_one_real ) ) ).

% inverse_eq_divide
thf(fact_7634_power__mult__power__inverse__commute,axiom,
    ! [X: complex,M: nat,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N2 ) @ ( power_power_complex @ X @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_7635_power__mult__power__inverse__commute,axiom,
    ! [X: real,M: nat,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N2 ) )
      = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N2 ) @ ( power_power_real @ X @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_7636_power__mult__inverse__distrib,axiom,
    ! [X: complex,M: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_7637_power__mult__inverse__distrib,axiom,
    ! [X: real,M: nat] :
      ( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
      = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_7638_mult__inverse__of__nat__commute,axiom,
    ! [Xa2: nat,X: complex] :
      ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa2 ) ) @ X )
      = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa2 ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_7639_mult__inverse__of__nat__commute,axiom,
    ! [Xa2: nat,X: real] :
      ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa2 ) ) @ X )
      = ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa2 ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_7640_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).

% nonzero_abs_inverse
thf(fact_7641_mult__inverse__of__int__commute,axiom,
    ! [Xa2: int,X: complex] :
      ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa2 ) ) @ X )
      = ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa2 ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_7642_mult__inverse__of__int__commute,axiom,
    ! [Xa2: int,X: real] :
      ( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa2 ) ) @ X )
      = ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa2 ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_7643_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_7644_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_7645_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P: nat > $o] :
        ? [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
          & ( P @ ( nat2 @ X2 ) ) ) ) ) ).

% ex_nat
thf(fact_7646_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P: nat > $o] :
        ! [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( P @ ( nat2 @ X2 ) ) ) ) ) ).

% all_nat
thf(fact_7647_eq__nat__nat__iff,axiom,
    ! [Z: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z8 ) )
          = ( Z = Z8 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_7648_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ B @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_7649_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_7650_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ B @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_7651_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_7652_inverse__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
      = ( ( ord_less_eq_real @ X @ zero_zero_real )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% inverse_le_1_iff
thf(fact_7653_one__less__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_less_inverse
thf(fact_7654_one__less__inverse__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
      = ( ( ord_less_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% one_less_inverse_iff
thf(fact_7655_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% field_class.field_inverse
thf(fact_7656_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% field_class.field_inverse
thf(fact_7657_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_7658_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_7659_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).

% inverse_add
thf(fact_7660_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% inverse_add
thf(fact_7661_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_7662_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_7663_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_7664_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_7665_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_7666_of__nat__ceiling,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_7667_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_7668_nat__le__iff,axiom,
    ! [X: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_7669_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_7670_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_7671_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_7672_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_7673_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_7674_cosh__def,axiom,
    ( cosh_real
    = ( ^ [X2: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).

% cosh_def
thf(fact_7675_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ B @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
         => ( ord_less_eq_real @ A @ B ) ) ) ) ).

% inverse_le_iff
thf(fact_7676_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ B @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
         => ( ord_less_real @ A @ B ) ) ) ) ).

% inverse_less_iff
thf(fact_7677_one__le__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_le_inverse
thf(fact_7678_inverse__less__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
      = ( ( ord_less_eq_real @ X @ zero_zero_real )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% inverse_less_1_iff
thf(fact_7679_one__le__inverse__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
      = ( ( ord_less_real @ zero_zero_real @ X )
        & ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% one_le_inverse_iff
thf(fact_7680_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_7681_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_7682_reals__Archimedean,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N3: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ X ) ) ).

% reals_Archimedean
thf(fact_7683_of__nat__floor,axiom,
    ! [R2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_7684_real__vector__eq__affinity,axiom,
    ! [M: real,Y: real,X: real,C: real] :
      ( ( M != zero_zero_real )
     => ( ( Y
          = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
        = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
          = X ) ) ) ).

% real_vector_eq_affinity
thf(fact_7685_real__vector__affinity__eq,axiom,
    ! [M: real,X: real,C: real,Y: real] :
      ( ( M != zero_zero_real )
     => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
          = Y )
        = ( X
          = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).

% real_vector_affinity_eq
thf(fact_7686_pos__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
        = ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_divideR_le_eq
thf(fact_7687_pos__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
        = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).

% pos_le_divideR_eq
thf(fact_7688_neg__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
        = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).

% neg_divideR_le_eq
thf(fact_7689_neg__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
        = ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_le_divideR_eq
thf(fact_7690_pos__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
        = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_divideR_less_eq
thf(fact_7691_pos__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).

% pos_less_divideR_eq
thf(fact_7692_neg__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).

% neg_divideR_less_eq
thf(fact_7693_neg__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
        = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_less_divideR_eq
thf(fact_7694_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_7695_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_7696_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_7697_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_7698_split__nat,axiom,
    ! [P2: nat > $o,I: int] :
      ( ( P2 @ ( nat2 @ I ) )
      = ( ! [N: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N ) )
           => ( P2 @ N ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_7699_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_7700_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_7701_nat__add__distrib,axiom,
    ! [Z: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z8 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_7702_nat__mult__distrib,axiom,
    ! [Z: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z8 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ).

% nat_mult_distrib
thf(fact_7703_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_7704_forall__pos__mono__1,axiom,
    ! [P2: real > $o,E2: real] :
      ( ! [D5: real,E: real] :
          ( ( ord_less_real @ D5 @ E )
         => ( ( P2 @ D5 )
           => ( P2 @ E ) ) )
     => ( ! [N3: nat] : ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P2 @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_7705_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N: nat] :
            ( ( N != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_7706_forall__pos__mono,axiom,
    ! [P2: real > $o,E2: real] :
      ( ! [D5: real,E: real] :
          ( ( ord_less_real @ D5 @ E )
         => ( ( P2 @ D5 )
           => ( P2 @ E ) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P2 @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_7707_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_7708_nat__diff__distrib,axiom,
    ! [Z8: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
     => ( ( ord_less_eq_int @ Z8 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z8 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_7709_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_7710_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_7711_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_7712_floor__eq3,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_7713_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_7714_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_7715_power__diff__conv__inverse,axiom,
    ! [X: complex,M: nat,N2: nat] :
      ( ( X != zero_zero_complex )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ M ) )
          = ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_7716_power__diff__conv__inverse,axiom,
    ! [X: real,M: nat,N2: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( power_power_real @ X @ ( minus_minus_nat @ N2 @ M ) )
          = ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_7717_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_7718_cosh__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) @ zero_zero_real )
      @ ( cosh_real @ X ) ) ).

% cosh_converges
thf(fact_7719_cosh__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X @ N ) ) @ zero_zero_complex )
      @ ( cosh_complex @ X ) ) ).

% cosh_converges
thf(fact_7720_pos__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divideR_eq
thf(fact_7721_pos__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_minus_divideR_le_eq
thf(fact_7722_neg__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_le_minus_divideR_eq
thf(fact_7723_neg__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divideR_le_eq
thf(fact_7724_pos__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divideR_eq
thf(fact_7725_pos__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_minus_divideR_less_eq
thf(fact_7726_neg__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_less_minus_divideR_eq
thf(fact_7727_neg__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divideR_less_eq
thf(fact_7728_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_7729_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_7730_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z8: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z8 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_7731_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_7732_floor__eq4,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_7733_diff__nat__eq__if,axiom,
    ! [Z8: int,Z: int] :
      ( ( ( ord_less_int @ Z8 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z8 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z8 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_7734_of__int__of__nat,axiom,
    ( ring_1_of_int_real
    = ( ^ [K2: int] : ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K2 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7735_of__int__of__nat,axiom,
    ( ring_1_of_int_int
    = ( ^ [K2: int] : ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K2 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K2 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7736_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_7737_cosh__field__def,axiom,
    ( cosh_real
    = ( ^ [Z6: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z6 ) @ ( exp_real @ ( uminus_uminus_real @ Z6 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_7738_exp__series__add__commuting,axiom,
    ! [X: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ N2 ) )
        = ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I5 ) ) @ ( power_power_complex @ X @ I5 ) ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ I5 ) ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ I5 ) ) ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% exp_series_add_commuting
thf(fact_7739_exp__series__add__commuting,axiom,
    ! [X: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ N2 ) )
        = ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I5 ) ) @ ( power_power_real @ X @ I5 ) ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ I5 ) ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ I5 ) ) ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% exp_series_add_commuting
thf(fact_7740_exp__first__term,axiom,
    ( exp_real
    = ( ^ [X2: real] :
          ( plus_plus_real @ one_one_real
          @ ( suminf_real
            @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_7741_exp__first__term,axiom,
    ( exp_complex
    = ( ^ [X2: complex] :
          ( plus_plus_complex @ one_one_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N ) ) ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_7742_tan__sec,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% tan_sec
thf(fact_7743_tan__sec,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% tan_sec
thf(fact_7744_sinh__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) )
      @ ( sinh_real @ X ) ) ).

% sinh_converges
thf(fact_7745_sinh__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X @ N ) ) )
      @ ( sinh_complex @ X ) ) ).

% sinh_converges
thf(fact_7746_cosh__double,axiom,
    ! [X: complex] :
      ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cosh_double
thf(fact_7747_cosh__double,axiom,
    ! [X: real] :
      ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cosh_double
thf(fact_7748_tanh__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( cosh_complex @ X )
       != zero_zero_complex )
     => ( ( ( cosh_complex @ Y )
         != zero_zero_complex )
       => ( ( tanh_complex @ ( plus_plus_complex @ X @ Y ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_7749_tanh__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( cosh_real @ X )
       != zero_zero_real )
     => ( ( ( cosh_real @ Y )
         != zero_zero_real )
       => ( ( tanh_real @ ( plus_plus_real @ X @ Y ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_7750_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I6: set_complex,F: complex > complex,I: complex] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_complex ) ) )
          @ F3 )
       => ( ( ( member_complex @ I @ I6 )
           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_complex @ I @ I6 )
           => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( groups808145749697022017omplex @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7751_sum__diff1_H__aux,axiom,
    ! [F3: set_Extended_enat,I6: set_Extended_enat,F: extended_enat > complex,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ord_le7203529160286727270d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I5: extended_enat] :
                ( ( member_Extended_enat @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_complex ) ) )
          @ F3 )
       => ( ( ( member_Extended_enat @ I @ I6 )
           => ( ( groups4395127934049735793omplex @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
              = ( minus_minus_complex @ ( groups4395127934049735793omplex @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_Extended_enat @ I @ I6 )
           => ( ( groups4395127934049735793omplex @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups4395127934049735793omplex @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7752_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I6: set_real,F: real > complex,I: real] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real2 @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_complex ) ) )
          @ F3 )
       => ( ( ( member_real2 @ I @ I6 )
           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
              = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_real2 @ I @ I6 )
           => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
              = ( groups5683813829254066239omplex @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7753_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I6: set_o,F: $o > complex,I: $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I5: $o] :
                ( ( member_o2 @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_complex ) ) )
          @ F3 )
       => ( ( ( member_o2 @ I @ I6 )
           => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
              = ( minus_minus_complex @ ( groups3443914341975893411omplex @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_o2 @ I @ I6 )
           => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
              = ( groups3443914341975893411omplex @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7754_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I6: set_int,F: int > complex,I: int] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int2 @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_complex ) ) )
          @ F3 )
       => ( ( ( member_int2 @ I @ I6 )
           => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
              = ( minus_minus_complex @ ( groups267424677133301183omplex @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_int2 @ I @ I6 )
           => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
              = ( groups267424677133301183omplex @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7755_sum__diff1_H__aux,axiom,
    ! [F3: set_complex,I6: set_complex,F: complex > int,I: complex] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_int ) ) )
          @ F3 )
       => ( ( ( member_complex @ I @ I6 )
           => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( minus_minus_int @ ( groups2909182065852811199ex_int @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_complex @ I @ I6 )
           => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
              = ( groups2909182065852811199ex_int @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7756_sum__diff1_H__aux,axiom,
    ! [F3: set_Extended_enat,I6: set_Extended_enat,F: extended_enat > int,I: extended_enat] :
      ( ( finite4001608067531595151d_enat @ F3 )
     => ( ( ord_le7203529160286727270d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [I5: extended_enat] :
                ( ( member_Extended_enat @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_int ) ) )
          @ F3 )
       => ( ( ( member_Extended_enat @ I @ I6 )
           => ( ( groups4847789403509879791at_int @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
              = ( minus_minus_int @ ( groups4847789403509879791at_int @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_Extended_enat @ I @ I6 )
           => ( ( groups4847789403509879791at_int @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
              = ( groups4847789403509879791at_int @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7757_sum__diff1_H__aux,axiom,
    ! [F3: set_real,I6: set_real,F: real > int,I: real] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real2 @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_int ) ) )
          @ F3 )
       => ( ( ( member_real2 @ I @ I6 )
           => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
              = ( minus_minus_int @ ( groups3901808747961969597al_int @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_real2 @ I @ I6 )
           => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
              = ( groups3901808747961969597al_int @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7758_sum__diff1_H__aux,axiom,
    ! [F3: set_o,I6: set_o,F: $o > int,I: $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [I5: $o] :
                ( ( member_o2 @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_int ) ) )
          @ F3 )
       => ( ( ( member_o2 @ I @ I6 )
           => ( ( groups4553916814277028129_o_int @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
              = ( minus_minus_int @ ( groups4553916814277028129_o_int @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_o2 @ I @ I6 )
           => ( ( groups4553916814277028129_o_int @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
              = ( groups4553916814277028129_o_int @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7759_sum__diff1_H__aux,axiom,
    ! [F3: set_int,I6: set_int,F: int > int,I: int] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int2 @ I5 @ I6 )
                & ( ( F @ I5 )
                 != zero_zero_int ) ) )
          @ F3 )
       => ( ( ( member_int2 @ I @ I6 )
           => ( ( groups2983280209131991357nt_int @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
              = ( minus_minus_int @ ( groups2983280209131991357nt_int @ F @ I6 ) @ ( F @ I ) ) ) )
          & ( ~ ( member_int2 @ I @ I6 )
           => ( ( groups2983280209131991357nt_int @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
              = ( groups2983280209131991357nt_int @ F @ I6 ) ) ) ) ) ) ).

% sum_diff1'_aux
thf(fact_7760_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_7761_or_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
      = A ) ).

% or.right_neutral
thf(fact_7762_or_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
      = A ) ).

% or.right_neutral
thf(fact_7763_or_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
      = A ) ).

% or.left_neutral
thf(fact_7764_or_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
      = A ) ).

% or.left_neutral
thf(fact_7765_sinh__0,axiom,
    ( ( sinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% sinh_0
thf(fact_7766_sinh__0,axiom,
    ( ( sinh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sinh_0
thf(fact_7767_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_7768_tanh__0,axiom,
    ( ( tanh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tanh_0
thf(fact_7769_sum_Oempty_H,axiom,
    ! [P4: real > nat] :
      ( ( groups3904299218471019873al_nat @ P4 @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty'
thf(fact_7770_sum_Oempty_H,axiom,
    ! [P4: real > real] :
      ( ( groups97945582718554045l_real @ P4 @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty'
thf(fact_7771_sum_Oempty_H,axiom,
    ! [P4: real > int] :
      ( ( groups3901808747961969597al_int @ P4 @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty'
thf(fact_7772_sum_Oempty_H,axiom,
    ! [P4: real > complex] :
      ( ( groups5683813829254066239omplex @ P4 @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty'
thf(fact_7773_sum_Oempty_H,axiom,
    ! [P4: real > extended_enat] :
      ( ( groups8946708780592123413d_enat @ P4 @ bot_bot_set_real )
      = zero_z5237406670263579293d_enat ) ).

% sum.empty'
thf(fact_7774_sum_Oempty_H,axiom,
    ! [P4: $o > nat] :
      ( ( groups4556407284786078405_o_nat @ P4 @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty'
thf(fact_7775_sum_Oempty_H,axiom,
    ! [P4: $o > real] :
      ( ( groups627172608727702305o_real @ P4 @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty'
thf(fact_7776_sum_Oempty_H,axiom,
    ! [P4: $o > int] :
      ( ( groups4553916814277028129_o_int @ P4 @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty'
thf(fact_7777_sum_Oempty_H,axiom,
    ! [P4: $o > complex] :
      ( ( groups3443914341975893411omplex @ P4 @ bot_bot_set_o )
      = zero_zero_complex ) ).

% sum.empty'
thf(fact_7778_sum_Oempty_H,axiom,
    ! [P4: $o > extended_enat] :
      ( ( groups8422889724626310577d_enat @ P4 @ bot_bot_set_o )
      = zero_z5237406670263579293d_enat ) ).

% sum.empty'
thf(fact_7779_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_7780_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_7781_sum_Oinsert_H,axiom,
    ! [I6: set_o,P4: $o > nat,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_o2 @ I @ I6 )
         => ( ( groups4556407284786078405_o_nat @ P4 @ ( insert_o2 @ I @ I6 ) )
            = ( groups4556407284786078405_o_nat @ P4 @ I6 ) ) )
        & ( ~ ( member_o2 @ I @ I6 )
         => ( ( groups4556407284786078405_o_nat @ P4 @ ( insert_o2 @ I @ I6 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups4556407284786078405_o_nat @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7782_sum_Oinsert_H,axiom,
    ! [I6: set_real,P4: real > nat,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_real2 @ I @ I6 )
         => ( ( groups3904299218471019873al_nat @ P4 @ ( insert_real2 @ I @ I6 ) )
            = ( groups3904299218471019873al_nat @ P4 @ I6 ) ) )
        & ( ~ ( member_real2 @ I @ I6 )
         => ( ( groups3904299218471019873al_nat @ P4 @ ( insert_real2 @ I @ I6 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups3904299218471019873al_nat @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7783_sum_Oinsert_H,axiom,
    ! [I6: set_nat,P4: nat > nat,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_nat2 @ I @ I6 )
         => ( ( groups1986416967739987077at_nat @ P4 @ ( insert_nat2 @ I @ I6 ) )
            = ( groups1986416967739987077at_nat @ P4 @ I6 ) ) )
        & ( ~ ( member_nat2 @ I @ I6 )
         => ( ( groups1986416967739987077at_nat @ P4 @ ( insert_nat2 @ I @ I6 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups1986416967739987077at_nat @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7784_sum_Oinsert_H,axiom,
    ! [I6: set_complex,P4: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_complex @ I @ I6 )
         => ( ( groups2911672536361861475ex_nat @ P4 @ ( insert_complex @ I @ I6 ) )
            = ( groups2911672536361861475ex_nat @ P4 @ I6 ) ) )
        & ( ~ ( member_complex @ I @ I6 )
         => ( ( groups2911672536361861475ex_nat @ P4 @ ( insert_complex @ I @ I6 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups2911672536361861475ex_nat @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7785_sum_Oinsert_H,axiom,
    ! [I6: set_int,P4: int > nat,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] :
              ( ( member_int2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_int2 @ I @ I6 )
         => ( ( groups2985770679641041633nt_nat @ P4 @ ( insert_int2 @ I @ I6 ) )
            = ( groups2985770679641041633nt_nat @ P4 @ I6 ) ) )
        & ( ~ ( member_int2 @ I @ I6 )
         => ( ( groups2985770679641041633nt_nat @ P4 @ ( insert_int2 @ I @ I6 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups2985770679641041633nt_nat @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7786_sum_Oinsert_H,axiom,
    ! [I6: set_Extended_enat,P4: extended_enat > nat,I: extended_enat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_Extended_enat @ I @ I6 )
         => ( ( groups4850279874018930067at_nat @ P4 @ ( insert_Extended_enat @ I @ I6 ) )
            = ( groups4850279874018930067at_nat @ P4 @ I6 ) ) )
        & ( ~ ( member_Extended_enat @ I @ I6 )
         => ( ( groups4850279874018930067at_nat @ P4 @ ( insert_Extended_enat @ I @ I6 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups4850279874018930067at_nat @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7787_sum_Oinsert_H,axiom,
    ! [I6: set_o,P4: $o > real,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_o2 @ I @ I6 )
         => ( ( groups627172608727702305o_real @ P4 @ ( insert_o2 @ I @ I6 ) )
            = ( groups627172608727702305o_real @ P4 @ I6 ) ) )
        & ( ~ ( member_o2 @ I @ I6 )
         => ( ( groups627172608727702305o_real @ P4 @ ( insert_o2 @ I @ I6 ) )
            = ( plus_plus_real @ ( P4 @ I ) @ ( groups627172608727702305o_real @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7788_sum_Oinsert_H,axiom,
    ! [I6: set_real,P4: real > real,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_real2 @ I @ I6 )
         => ( ( groups97945582718554045l_real @ P4 @ ( insert_real2 @ I @ I6 ) )
            = ( groups97945582718554045l_real @ P4 @ I6 ) ) )
        & ( ~ ( member_real2 @ I @ I6 )
         => ( ( groups97945582718554045l_real @ P4 @ ( insert_real2 @ I @ I6 ) )
            = ( plus_plus_real @ ( P4 @ I ) @ ( groups97945582718554045l_real @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7789_sum_Oinsert_H,axiom,
    ! [I6: set_nat,P4: nat > real,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat2 @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_nat2 @ I @ I6 )
         => ( ( groups8560362682196896993t_real @ P4 @ ( insert_nat2 @ I @ I6 ) )
            = ( groups8560362682196896993t_real @ P4 @ I6 ) ) )
        & ( ~ ( member_nat2 @ I @ I6 )
         => ( ( groups8560362682196896993t_real @ P4 @ ( insert_nat2 @ I @ I6 ) )
            = ( plus_plus_real @ ( P4 @ I ) @ ( groups8560362682196896993t_real @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7790_sum_Oinsert_H,axiom,
    ! [I6: set_complex,P4: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ I6 )
              & ( ( P4 @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( ( member_complex @ I @ I6 )
         => ( ( groups5737402329758386879x_real @ P4 @ ( insert_complex @ I @ I6 ) )
            = ( groups5737402329758386879x_real @ P4 @ I6 ) ) )
        & ( ~ ( member_complex @ I @ I6 )
         => ( ( groups5737402329758386879x_real @ P4 @ ( insert_complex @ I @ I6 ) )
            = ( plus_plus_real @ ( P4 @ I ) @ ( groups5737402329758386879x_real @ P4 @ I6 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_7791_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_7792_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_7793_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_7794_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_7795_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_7796_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_7797_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_7798_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_7799_bit_Odisj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
      = X ) ).

% bit.disj_zero_right
thf(fact_7800_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se1409905431419307370or_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
thf(fact_7801_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( bit_se1412395901928357646or_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
thf(fact_7802_sum_Onon__neutral_H,axiom,
    ! [G: $o > nat,I6: set_o] :
      ( ( groups4556407284786078405_o_nat @ G
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
      = ( groups4556407284786078405_o_nat @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7803_sum_Onon__neutral_H,axiom,
    ! [G: real > nat,I6: set_real] :
      ( ( groups3904299218471019873al_nat @ G
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
      = ( groups3904299218471019873al_nat @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7804_sum_Onon__neutral_H,axiom,
    ! [G: nat > nat,I6: set_nat] :
      ( ( groups1986416967739987077at_nat @ G
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
      = ( groups1986416967739987077at_nat @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7805_sum_Onon__neutral_H,axiom,
    ! [G: int > nat,I6: set_int] :
      ( ( groups2985770679641041633nt_nat @ G
        @ ( collect_int
          @ ^ [X2: int] :
              ( ( member_int2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
      = ( groups2985770679641041633nt_nat @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7806_sum_Onon__neutral_H,axiom,
    ! [G: $o > real,I6: set_o] :
      ( ( groups627172608727702305o_real @ G
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
      = ( groups627172608727702305o_real @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7807_sum_Onon__neutral_H,axiom,
    ! [G: real > real,I6: set_real] :
      ( ( groups97945582718554045l_real @ G
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
      = ( groups97945582718554045l_real @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7808_sum_Onon__neutral_H,axiom,
    ! [G: nat > real,I6: set_nat] :
      ( ( groups8560362682196896993t_real @ G
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
      = ( groups8560362682196896993t_real @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7809_sum_Onon__neutral_H,axiom,
    ! [G: int > real,I6: set_int] :
      ( ( groups1523912220035142973t_real @ G
        @ ( collect_int
          @ ^ [X2: int] :
              ( ( member_int2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
      = ( groups1523912220035142973t_real @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7810_sum_Onon__neutral_H,axiom,
    ! [G: $o > int,I6: set_o] :
      ( ( groups4553916814277028129_o_int @ G
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_int ) ) ) )
      = ( groups4553916814277028129_o_int @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7811_sum_Onon__neutral_H,axiom,
    ! [G: real > int,I6: set_real] :
      ( ( groups3901808747961969597al_int @ G
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_int ) ) ) )
      = ( groups3901808747961969597al_int @ G @ I6 ) ) ).

% sum.non_neutral'
thf(fact_7812_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N3 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N3 ) )
     => ( ( plus_plus_int @ A @ B )
        = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_7813_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N3 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N3 ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_7814_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: nat > nat,H2: nat > nat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( groups1986416967739987077at_nat
          @ ^ [I5: nat] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I6 ) @ ( groups1986416967739987077at_nat @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7815_sum_Odistrib__triv_H,axiom,
    ! [I6: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( groups2911672536361861475ex_nat
          @ ^ [I5: complex] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_nat @ ( groups2911672536361861475ex_nat @ G @ I6 ) @ ( groups2911672536361861475ex_nat @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7816_sum_Odistrib__triv_H,axiom,
    ! [I6: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( groups2985770679641041633nt_nat
          @ ^ [I5: int] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I6 ) @ ( groups2985770679641041633nt_nat @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7817_sum_Odistrib__triv_H,axiom,
    ! [I6: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( groups4850279874018930067at_nat
          @ ^ [I5: extended_enat] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_nat @ ( groups4850279874018930067at_nat @ G @ I6 ) @ ( groups4850279874018930067at_nat @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7818_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: nat > int,H2: nat > int] :
      ( ( finite_finite_nat @ I6 )
     => ( ( groups1983926497230936801at_int
          @ ^ [I5: nat] : ( plus_plus_int @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_int @ ( groups1983926497230936801at_int @ G @ I6 ) @ ( groups1983926497230936801at_int @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7819_sum_Odistrib__triv_H,axiom,
    ! [I6: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( groups2909182065852811199ex_int
          @ ^ [I5: complex] : ( plus_plus_int @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_int @ ( groups2909182065852811199ex_int @ G @ I6 ) @ ( groups2909182065852811199ex_int @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7820_sum_Odistrib__triv_H,axiom,
    ! [I6: set_int,G: int > int,H2: int > int] :
      ( ( finite_finite_int @ I6 )
     => ( ( groups2983280209131991357nt_int
          @ ^ [I5: int] : ( plus_plus_int @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_int @ ( groups2983280209131991357nt_int @ G @ I6 ) @ ( groups2983280209131991357nt_int @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7821_sum_Odistrib__triv_H,axiom,
    ! [I6: set_Extended_enat,G: extended_enat > int,H2: extended_enat > int] :
      ( ( finite4001608067531595151d_enat @ I6 )
     => ( ( groups4847789403509879791at_int
          @ ^ [I5: extended_enat] : ( plus_plus_int @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_int @ ( groups4847789403509879791at_int @ G @ I6 ) @ ( groups4847789403509879791at_int @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7822_sum_Odistrib__triv_H,axiom,
    ! [I6: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( groups8560362682196896993t_real
          @ ^ [I5: nat] : ( plus_plus_real @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I6 ) @ ( groups8560362682196896993t_real @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7823_sum_Odistrib__triv_H,axiom,
    ! [I6: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( groups5737402329758386879x_real
          @ ^ [I5: complex] : ( plus_plus_real @ ( G @ I5 ) @ ( H2 @ I5 ) )
          @ I6 )
        = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I6 ) @ ( groups5737402329758386879x_real @ H2 @ I6 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_7824_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > nat,H2: real > nat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_nat ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups3904299218471019873al_nat @ G @ T3 )
            = ( groups3904299218471019873al_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7825_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_o,T3: set_o,G: $o > nat,H2: $o > nat] :
      ( ( ord_less_eq_set_o @ S3 @ T3 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_nat ) )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups4556407284786078405_o_nat @ G @ T3 )
            = ( groups4556407284786078405_o_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7826_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > nat,H2: int > nat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_nat ) )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups2985770679641041633nt_nat @ G @ T3 )
            = ( groups2985770679641041633nt_nat @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7827_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > real,H2: real > real] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_real ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups97945582718554045l_real @ G @ T3 )
            = ( groups97945582718554045l_real @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7828_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_o,T3: set_o,G: $o > real,H2: $o > real] :
      ( ( ord_less_eq_set_o @ S3 @ T3 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_real ) )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups627172608727702305o_real @ G @ T3 )
            = ( groups627172608727702305o_real @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7829_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > real,H2: int > real] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_real ) )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups1523912220035142973t_real @ G @ T3 )
            = ( groups1523912220035142973t_real @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7830_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > int,H2: real > int] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_int ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups3901808747961969597al_int @ G @ T3 )
            = ( groups3901808747961969597al_int @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7831_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_o,T3: set_o,G: $o > int,H2: $o > int] :
      ( ( ord_less_eq_set_o @ S3 @ T3 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_int ) )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups4553916814277028129_o_int @ G @ T3 )
            = ( groups4553916814277028129_o_int @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7832_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_int,T3: set_int,G: int > int,H2: int > int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_int ) )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups2983280209131991357nt_int @ G @ T3 )
            = ( groups2983280209131991357nt_int @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7833_sum_Omono__neutral__cong__right_H,axiom,
    ! [S3: set_real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_complex ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups5683813829254066239omplex @ G @ T3 )
            = ( groups5683813829254066239omplex @ H2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
thf(fact_7834_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > nat,G: real > nat] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_nat ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups3904299218471019873al_nat @ G @ S3 )
            = ( groups3904299218471019873al_nat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7835_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_o,T3: set_o,H2: $o > nat,G: $o > nat] :
      ( ( ord_less_eq_set_o @ S3 @ T3 )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ ( minus_minus_set_o @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_nat ) )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups4556407284786078405_o_nat @ G @ S3 )
            = ( groups4556407284786078405_o_nat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7836_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_int,T3: set_int,H2: int > nat,G: int > nat] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_nat ) )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups2985770679641041633nt_nat @ G @ S3 )
            = ( groups2985770679641041633nt_nat @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7837_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > real,G: real > real] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_real ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups97945582718554045l_real @ G @ S3 )
            = ( groups97945582718554045l_real @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7838_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_o,T3: set_o,H2: $o > real,G: $o > real] :
      ( ( ord_less_eq_set_o @ S3 @ T3 )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ ( minus_minus_set_o @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_real ) )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups627172608727702305o_real @ G @ S3 )
            = ( groups627172608727702305o_real @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7839_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_int,T3: set_int,H2: int > real,G: int > real] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_real ) )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups1523912220035142973t_real @ G @ S3 )
            = ( groups1523912220035142973t_real @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7840_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > int,G: real > int] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_int ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups3901808747961969597al_int @ G @ S3 )
            = ( groups3901808747961969597al_int @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7841_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_o,T3: set_o,H2: $o > int,G: $o > int] :
      ( ( ord_less_eq_set_o @ S3 @ T3 )
     => ( ! [I3: $o] :
            ( ( member_o2 @ I3 @ ( minus_minus_set_o @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_int ) )
       => ( ! [X5: $o] :
              ( ( member_o2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups4553916814277028129_o_int @ G @ S3 )
            = ( groups4553916814277028129_o_int @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7842_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_int,T3: set_int,H2: int > int,G: int > int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ! [I3: int] :
            ( ( member_int2 @ I3 @ ( minus_minus_set_int @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_int ) )
       => ( ! [X5: int] :
              ( ( member_int2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups2983280209131991357nt_int @ G @ S3 )
            = ( groups2983280209131991357nt_int @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7843_sum_Omono__neutral__cong__left_H,axiom,
    ! [S3: set_real,T3: set_real,H2: real > complex,G: real > complex] :
      ( ( ord_less_eq_set_real @ S3 @ T3 )
     => ( ! [I3: real] :
            ( ( member_real2 @ I3 @ ( minus_minus_set_real @ T3 @ S3 ) )
           => ( ( H2 @ I3 )
              = zero_zero_complex ) )
       => ( ! [X5: real] :
              ( ( member_real2 @ X5 @ S3 )
             => ( ( G @ X5 )
                = ( H2 @ X5 ) ) )
         => ( ( groups5683813829254066239omplex @ G @ S3 )
            = ( groups5683813829254066239omplex @ H2 @ T3 ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
thf(fact_7844_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_nat ) )
       => ( ( groups1986416967739987077at_nat @ G @ T3 )
          = ( groups1986416967739987077at_nat @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_7845_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_real ) )
       => ( ( groups8560362682196896993t_real @ G @ T3 )
          = ( groups8560362682196896993t_real @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_7846_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_int ) )
       => ( ( groups1983926497230936801at_int @ G @ T3 )
          = ( groups1983926497230936801at_int @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_7847_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > complex] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_complex ) )
       => ( ( groups8515261248781899619omplex @ G @ T3 )
          = ( groups8515261248781899619omplex @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_7848_sum_Omono__neutral__right_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > extended_enat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_z5237406670263579293d_enat ) )
       => ( ( groups707763781290628081d_enat @ G @ T3 )
          = ( groups707763781290628081d_enat @ G @ S3 ) ) ) ) ).

% sum.mono_neutral_right'
thf(fact_7849_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_nat ) )
       => ( ( groups1986416967739987077at_nat @ G @ S3 )
          = ( groups1986416967739987077at_nat @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_7850_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_real ) )
       => ( ( groups8560362682196896993t_real @ G @ S3 )
          = ( groups8560362682196896993t_real @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_7851_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_int ) )
       => ( ( groups1983926497230936801at_int @ G @ S3 )
          = ( groups1983926497230936801at_int @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_7852_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > complex] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_zero_complex ) )
       => ( ( groups8515261248781899619omplex @ G @ S3 )
          = ( groups8515261248781899619omplex @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_7853_sum_Omono__neutral__left_H,axiom,
    ! [S3: set_nat,T3: set_nat,G: nat > extended_enat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
           => ( ( G @ X5 )
              = zero_z5237406670263579293d_enat ) )
       => ( ( groups707763781290628081d_enat @ G @ S3 )
          = ( groups707763781290628081d_enat @ G @ T3 ) ) ) ) ).

% sum.mono_neutral_left'
thf(fact_7854_sinh__add,axiom,
    ! [X: real,Y: real] :
      ( ( sinh_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).

% sinh_add
thf(fact_7855_sinh__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( sinh_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ Y ) ) ) ) ).

% sinh_add
thf(fact_7856_cosh__add,axiom,
    ! [X: real,Y: real] :
      ( ( cosh_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).

% cosh_add
thf(fact_7857_cosh__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( cosh_complex @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cosh_complex @ X ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( sinh_complex @ X ) @ ( sinh_complex @ Y ) ) ) ) ).

% cosh_add
thf(fact_7858_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ X )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ A @ X )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
            = zero_zero_int )
         => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
              = ( uminus_uminus_int @ one_one_int ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_7859_sinh__plus__cosh,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
      = ( exp_real @ X ) ) ).

% sinh_plus_cosh
thf(fact_7860_cosh__plus__sinh,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
      = ( exp_real @ X ) ) ).

% cosh_plus_sinh
thf(fact_7861_sum_Odistrib_H,axiom,
    ! [I6: set_o,G: $o > nat,H2: $o > nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups4556407284786078405_o_nat
            @ ^ [I5: $o] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_nat @ ( groups4556407284786078405_o_nat @ G @ I6 ) @ ( groups4556407284786078405_o_nat @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7862_sum_Odistrib_H,axiom,
    ! [I6: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups3904299218471019873al_nat
            @ ^ [I5: real] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_nat @ ( groups3904299218471019873al_nat @ G @ I6 ) @ ( groups3904299218471019873al_nat @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7863_sum_Odistrib_H,axiom,
    ! [I6: set_nat,G: nat > nat,H2: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups1986416967739987077at_nat
            @ ^ [I5: nat] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I6 ) @ ( groups1986416967739987077at_nat @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7864_sum_Odistrib_H,axiom,
    ! [I6: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups2911672536361861475ex_nat
            @ ^ [I5: complex] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_nat @ ( groups2911672536361861475ex_nat @ G @ I6 ) @ ( groups2911672536361861475ex_nat @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7865_sum_Odistrib_H,axiom,
    ! [I6: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] :
              ( ( member_int2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups2985770679641041633nt_nat
            @ ^ [I5: int] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I6 ) @ ( groups2985770679641041633nt_nat @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7866_sum_Odistrib_H,axiom,
    ! [I6: set_Extended_enat,G: extended_enat > nat,H2: extended_enat > nat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [X2: extended_enat] :
              ( ( member_Extended_enat @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_nat ) ) ) )
     => ( ( finite4001608067531595151d_enat
          @ ( collec4429806609662206161d_enat
            @ ^ [X2: extended_enat] :
                ( ( member_Extended_enat @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups4850279874018930067at_nat
            @ ^ [I5: extended_enat] : ( plus_plus_nat @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_nat @ ( groups4850279874018930067at_nat @ G @ I6 ) @ ( groups4850279874018930067at_nat @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7867_sum_Odistrib_H,axiom,
    ! [I6: set_o,G: $o > real,H2: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X2: $o] :
              ( ( member_o2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_real ) ) ) )
       => ( ( groups627172608727702305o_real
            @ ^ [I5: $o] : ( plus_plus_real @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_real @ ( groups627172608727702305o_real @ G @ I6 ) @ ( groups627172608727702305o_real @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7868_sum_Odistrib_H,axiom,
    ! [I6: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( member_real2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_real ) ) ) )
       => ( ( groups97945582718554045l_real
            @ ^ [I5: real] : ( plus_plus_real @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_real @ ( groups97945582718554045l_real @ G @ I6 ) @ ( groups97945582718554045l_real @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7869_sum_Odistrib_H,axiom,
    ! [I6: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat2 @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat2 @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_real ) ) ) )
       => ( ( groups8560362682196896993t_real
            @ ^ [I5: nat] : ( plus_plus_real @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_real @ ( groups8560362682196896993t_real @ G @ I6 ) @ ( groups8560362682196896993t_real @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7870_sum_Odistrib_H,axiom,
    ! [I6: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ I6 )
              & ( ( G @ X2 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ I6 )
                & ( ( H2 @ X2 )
                 != zero_zero_real ) ) ) )
       => ( ( groups5737402329758386879x_real
            @ ^ [I5: complex] : ( plus_plus_real @ ( G @ I5 ) @ ( H2 @ I5 ) )
            @ I6 )
          = ( plus_plus_real @ ( groups5737402329758386879x_real @ G @ I6 ) @ ( groups5737402329758386879x_real @ H2 @ I6 ) ) ) ) ) ).

% sum.distrib'
thf(fact_7871_sum_OG__def,axiom,
    ( groups4556407284786078405_o_nat
    = ( ^ [P6: $o > nat,I7: set_o] :
          ( if_nat
          @ ( finite_finite_o
            @ ( collect_o
              @ ^ [X2: $o] :
                  ( ( member_o2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ ( groups8507830703676809646_o_nat @ P6
            @ ( collect_o
              @ ^ [X2: $o] :
                  ( ( member_o2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_7872_sum_OG__def,axiom,
    ( groups3904299218471019873al_nat
    = ( ^ [P6: real > nat,I7: set_real] :
          ( if_nat
          @ ( finite_finite_real
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( member_real2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ ( groups1935376822645274424al_nat @ P6
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( member_real2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_7873_sum_OG__def,axiom,
    ( groups2911672536361861475ex_nat
    = ( ^ [P6: complex > nat,I7: set_complex] :
          ( if_nat
          @ ( finite3207457112153483333omplex
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( member_complex @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ ( groups5693394587270226106ex_nat @ P6
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( member_complex @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_7874_sum_OG__def,axiom,
    ( groups2985770679641041633nt_nat
    = ( ^ [P6: int > nat,I7: set_int] :
          ( if_nat
          @ ( finite_finite_int
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( member_int2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ ( groups4541462559716669496nt_nat @ P6
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( member_int2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_7875_sum_OG__def,axiom,
    ( groups4850279874018930067at_nat
    = ( ^ [P6: extended_enat > nat,I7: set_Extended_enat] :
          ( if_nat
          @ ( finite4001608067531595151d_enat
            @ ( collec4429806609662206161d_enat
              @ ^ [X2: extended_enat] :
                  ( ( member_Extended_enat @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ ( groups2027974829824023292at_nat @ P6
            @ ( collec4429806609662206161d_enat
              @ ^ [X2: extended_enat] :
                  ( ( member_Extended_enat @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_7876_sum_OG__def,axiom,
    ( groups627172608727702305o_real
    = ( ^ [P6: $o > real,I7: set_o] :
          ( if_real
          @ ( finite_finite_o
            @ ( collect_o
              @ ^ [X2: $o] :
                  ( ( member_o2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ ( groups8691415230153176458o_real @ P6
            @ ( collect_o
              @ ^ [X2: $o] :
                  ( ( member_o2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ zero_zero_real ) ) ) ).

% sum.G_def
thf(fact_7877_sum_OG__def,axiom,
    ( groups97945582718554045l_real
    = ( ^ [P6: real > real,I7: set_real] :
          ( if_real
          @ ( finite_finite_real
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( member_real2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ ( groups8097168146408367636l_real @ P6
            @ ( collect_real
              @ ^ [X2: real] :
                  ( ( member_real2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ zero_zero_real ) ) ) ).

% sum.G_def
thf(fact_7878_sum_OG__def,axiom,
    ( groups5737402329758386879x_real
    = ( ^ [P6: complex > real,I7: set_complex] :
          ( if_real
          @ ( finite3207457112153483333omplex
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( member_complex @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ ( groups5808333547571424918x_real @ P6
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( member_complex @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ zero_zero_real ) ) ) ).

% sum.G_def
thf(fact_7879_sum_OG__def,axiom,
    ( groups1523912220035142973t_real
    = ( ^ [P6: int > real,I7: set_int] :
          ( if_real
          @ ( finite_finite_int
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( member_int2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ ( groups8778361861064173332t_real @ P6
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( member_int2 @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ zero_zero_real ) ) ) ).

% sum.G_def
thf(fact_7880_sum_OG__def,axiom,
    ( groups1070518202123951855t_real
    = ( ^ [P6: extended_enat > real,I7: set_Extended_enat] :
          ( if_real
          @ ( finite4001608067531595151d_enat
            @ ( collec4429806609662206161d_enat
              @ ^ [X2: extended_enat] :
                  ( ( member_Extended_enat @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ ( groups4148127829035722712t_real @ P6
            @ ( collec4429806609662206161d_enat
              @ ^ [X2: extended_enat] :
                  ( ( member_Extended_enat @ X2 @ I7 )
                  & ( ( P6 @ X2 )
                   != zero_zero_real ) ) ) )
          @ zero_zero_real ) ) ) ).

% sum.G_def
thf(fact_7881_mask__Suc__exp,axiom,
    ! [N2: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N2 ) )
      = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).

% mask_Suc_exp
thf(fact_7882_mask__Suc__exp,axiom,
    ! [N2: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N2 ) )
      = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% mask_Suc_exp
thf(fact_7883_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_7884_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_7885_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_7886_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_7887_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N2 ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_7888_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N2 ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_7889_sinh__zero__iff,axiom,
    ! [X: complex] :
      ( ( ( sinh_complex @ X )
        = zero_zero_complex )
      = ( member_complex @ ( exp_complex @ X ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).

% sinh_zero_iff
thf(fact_7890_sinh__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( member_real2 @ ( exp_real @ X ) @ ( insert_real2 @ one_one_real @ ( insert_real2 @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).

% sinh_zero_iff
thf(fact_7891_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7892_sum__diff1_H,axiom,
    ! [I6: set_complex,F: complex > complex,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_complex @ I @ I6 )
         => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( minus_minus_complex @ ( groups808145749697022017omplex @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_complex @ I @ I6 )
         => ( ( groups808145749697022017omplex @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( groups808145749697022017omplex @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7893_sum__diff1_H,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > complex,I: extended_enat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I5: extended_enat] :
              ( ( member_Extended_enat @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_Extended_enat @ I @ I6 )
         => ( ( groups4395127934049735793omplex @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_complex @ ( groups4395127934049735793omplex @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_Extended_enat @ I @ I6 )
         => ( ( groups4395127934049735793omplex @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups4395127934049735793omplex @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7894_sum__diff1_H,axiom,
    ! [I6: set_real,F: real > complex,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real2 @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_real2 @ I @ I6 )
         => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
            = ( minus_minus_complex @ ( groups5683813829254066239omplex @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_real2 @ I @ I6 )
         => ( ( groups5683813829254066239omplex @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
            = ( groups5683813829254066239omplex @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7895_sum__diff1_H,axiom,
    ! [I6: set_o,F: $o > complex,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I5: $o] :
              ( ( member_o2 @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_o2 @ I @ I6 )
         => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
            = ( minus_minus_complex @ ( groups3443914341975893411omplex @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_o2 @ I @ I6 )
         => ( ( groups3443914341975893411omplex @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
            = ( groups3443914341975893411omplex @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7896_sum__diff1_H,axiom,
    ! [I6: set_int,F: int > complex,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int2 @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( ( member_int2 @ I @ I6 )
         => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
            = ( minus_minus_complex @ ( groups267424677133301183omplex @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_int2 @ I @ I6 )
         => ( ( groups267424677133301183omplex @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
            = ( groups267424677133301183omplex @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7897_sum__diff1_H,axiom,
    ! [I6: set_complex,F: complex > int,I: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_int ) ) ) )
     => ( ( ( member_complex @ I @ I6 )
         => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups2909182065852811199ex_int @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_complex @ I @ I6 )
         => ( ( groups2909182065852811199ex_int @ F @ ( minus_811609699411566653omplex @ I6 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
            = ( groups2909182065852811199ex_int @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7898_sum__diff1_H,axiom,
    ! [I6: set_Extended_enat,F: extended_enat > int,I: extended_enat] :
      ( ( finite4001608067531595151d_enat
        @ ( collec4429806609662206161d_enat
          @ ^ [I5: extended_enat] :
              ( ( member_Extended_enat @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_int ) ) ) )
     => ( ( ( member_Extended_enat @ I @ I6 )
         => ( ( groups4847789403509879791at_int @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
            = ( minus_minus_int @ ( groups4847789403509879791at_int @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_Extended_enat @ I @ I6 )
         => ( ( groups4847789403509879791at_int @ F @ ( minus_925952699566721837d_enat @ I6 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
            = ( groups4847789403509879791at_int @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7899_sum__diff1_H,axiom,
    ! [I6: set_real,F: real > int,I: real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real2 @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_int ) ) ) )
     => ( ( ( member_real2 @ I @ I6 )
         => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
            = ( minus_minus_int @ ( groups3901808747961969597al_int @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_real2 @ I @ I6 )
         => ( ( groups3901808747961969597al_int @ F @ ( minus_minus_set_real @ I6 @ ( insert_real2 @ I @ bot_bot_set_real ) ) )
            = ( groups3901808747961969597al_int @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7900_sum__diff1_H,axiom,
    ! [I6: set_o,F: $o > int,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I5: $o] :
              ( ( member_o2 @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_int ) ) ) )
     => ( ( ( member_o2 @ I @ I6 )
         => ( ( groups4553916814277028129_o_int @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
            = ( minus_minus_int @ ( groups4553916814277028129_o_int @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_o2 @ I @ I6 )
         => ( ( groups4553916814277028129_o_int @ F @ ( minus_minus_set_o @ I6 @ ( insert_o2 @ I @ bot_bot_set_o ) ) )
            = ( groups4553916814277028129_o_int @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7901_sum__diff1_H,axiom,
    ! [I6: set_int,F: int > int,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int2 @ I5 @ I6 )
              & ( ( F @ I5 )
               != zero_zero_int ) ) ) )
     => ( ( ( member_int2 @ I @ I6 )
         => ( ( groups2983280209131991357nt_int @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
            = ( minus_minus_int @ ( groups2983280209131991357nt_int @ F @ I6 ) @ ( F @ I ) ) ) )
        & ( ~ ( member_int2 @ I @ I6 )
         => ( ( groups2983280209131991357nt_int @ F @ ( minus_minus_set_int @ I6 @ ( insert_int2 @ I @ bot_bot_set_int ) ) )
            = ( groups2983280209131991357nt_int @ F @ I6 ) ) ) ) ) ).

% sum_diff1'
thf(fact_7902_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_7903_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_7904_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_7905_cosh__square__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).

% cosh_square_eq
thf(fact_7906_cosh__square__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).

% cosh_square_eq
thf(fact_7907_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_7908_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M2: nat] :
          ( ( ord_less_eq_nat @ M9 @ M2 )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M9 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M2 ) @ ( X8 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_7909_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_7910_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A4: set_real] :
          ( A4
          = ( insert_real2 @ ( the_elem_real @ A4 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7911_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
          ( A4
          = ( insert_o2 @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7912_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A4: set_nat] :
          ( A4
          = ( insert_nat2 @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7913_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A4: set_int] :
          ( A4
          = ( insert_int2 @ ( the_elem_int @ A4 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_7914_length__subseqs,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( size_s8217280938318005548T_VEBT @ ( subseqs_VEBT_VEBT @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_subseqs
thf(fact_7915_length__subseqs,axiom,
    ! [Xs: list_int] :
      ( ( size_s533118279054570080st_int @ ( subseqs_int @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_int @ Xs ) ) ) ).

% length_subseqs
thf(fact_7916_length__subseqs,axiom,
    ! [Xs: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( subseqs_nat @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_subseqs
thf(fact_7917_atLeastLessThan__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o2 @ I @ ( set_or7139685690850216873Than_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_o @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7918_atLeastLessThan__iff,axiom,
    ! [I: extended_enat,L: extended_enat,U: extended_enat] :
      ( ( member_Extended_enat @ I @ ( set_or4374356025156299511d_enat @ L @ U ) )
      = ( ( ord_le2932123472753598470d_enat @ L @ I )
        & ( ord_le72135733267957522d_enat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7919_atLeastLessThan__iff,axiom,
    ! [I: filter_nat,L: filter_nat,U: filter_nat] :
      ( ( member_filter_nat @ I @ ( set_or1773934645810362255er_nat @ L @ U ) )
      = ( ( ord_le2510731241096832064er_nat @ L @ I )
        & ( ord_less_filter_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7920_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real2 @ I @ ( set_or66887138388493659n_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_real @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7921_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat2 @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_set_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7922_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat2 @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7923_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int2 @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_int @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_7924_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_eq_o @ B @ A )
     => ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
thf(fact_7925_atLeastLessThan__empty,axiom,
    ! [B: filter_nat,A: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ B @ A )
     => ( ( set_or1773934645810362255er_nat @ A @ B )
        = bot_bo498966703094740906er_nat ) ) ).

% atLeastLessThan_empty
thf(fact_7926_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( set_or66887138388493659n_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
thf(fact_7927_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( set_or3540276404033026485et_nat @ A @ B )
        = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_7928_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_7929_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_7930_ivl__subset,axiom,
    ! [I: real,J: real,M: real,N2: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J ) @ ( set_or66887138388493659n_real @ M @ N2 ) )
      = ( ( ord_less_eq_real @ J @ I )
        | ( ( ord_less_eq_real @ M @ I )
          & ( ord_less_eq_real @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7931_ivl__subset,axiom,
    ! [I: nat,J: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ J @ I )
        | ( ( ord_less_eq_nat @ M @ I )
          & ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7932_ivl__subset,axiom,
    ! [I: int,J: int,M: int,N2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N2 ) )
      = ( ( ord_less_eq_int @ J @ I )
        | ( ( ord_less_eq_int @ M @ I )
          & ( ord_less_eq_int @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_7933_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_7934_atLeastLessThan__empty__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( set_or4374356025156299511d_enat @ A @ B )
        = bot_bo7653980558646680370d_enat )
      = ( ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_7935_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_real @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_7936_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_7937_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_7938_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or7139685690850216873Than_o @ A @ B ) )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_7939_atLeastLessThan__empty__iff2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat
        = ( set_or4374356025156299511d_enat @ A @ B ) )
      = ( ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_7940_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or66887138388493659n_real @ A @ B ) )
      = ( ~ ( ord_less_real @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_7941_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_7942_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A @ B ) )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_7943_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_7944_ivl__diff,axiom,
    ! [I: real,N2: real,M: real] :
      ( ( ord_less_eq_real @ I @ N2 )
     => ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I @ M ) @ ( set_or66887138388493659n_real @ I @ N2 ) )
        = ( set_or66887138388493659n_real @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_7945_ivl__diff,axiom,
    ! [I: nat,N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N2 ) )
        = ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_7946_ivl__diff,axiom,
    ! [I: int,N2: int,M: int] :
      ( ( ord_less_eq_int @ I @ N2 )
     => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N2 ) )
        = ( set_or4662586982721622107an_int @ N2 @ M ) ) ) ).

% ivl_diff
thf(fact_7947_is__singletonI,axiom,
    ! [X: real] : ( is_singleton_real @ ( insert_real2 @ X @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_7948_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o2 @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_7949_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_7950_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int2 @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_7951_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat2 @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_7952_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_7953_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_7954_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_7955_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_7956_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_7957_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_7958_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_7959_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_on7984719198319812577d_enat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_7960_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_7961_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_7962_atLeastLessThan__eq__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ C @ D )
       => ( ( ( set_or4374356025156299511d_enat @ A @ B )
            = ( set_or4374356025156299511d_enat @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7963_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ( set_or66887138388493659n_real @ A @ B )
            = ( set_or66887138388493659n_real @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7964_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7965_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C @ D ) )
          = ( ( A = C )
            & ( B = D ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_7966_atLeastLessThan__inj_I1_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ( set_or4374356025156299511d_enat @ A @ B )
        = ( set_or4374356025156299511d_enat @ C @ D ) )
     => ( ( ord_le72135733267957522d_enat @ A @ B )
       => ( ( ord_le72135733267957522d_enat @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7967_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7968_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7969_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_7970_atLeastLessThan__inj_I2_J,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ( set_or4374356025156299511d_enat @ A @ B )
        = ( set_or4374356025156299511d_enat @ C @ D ) )
     => ( ( ord_le72135733267957522d_enat @ A @ B )
       => ( ( ord_le72135733267957522d_enat @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7971_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7972_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7973_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D )
         => ( B = D ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_7974_atLeastLessThan__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
     => ( ( ord_less_eq_real @ B @ A )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7975_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7976_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_7977_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_7978_all__nat__less__eq,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
           => ( P2 @ M2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P2 @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_7979_ex__nat__less__eq,axiom,
    ! [N2: nat,P2: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
            & ( P2 @ M2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P2 @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_7980_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_7981_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_7982_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_7983_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_7984_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_7985_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_7986_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_7987_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_7988_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_7989_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_7990_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_7991_is__singletonI_H,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ! [X5: set_nat,Y3: set_nat] :
            ( ( member_set_nat2 @ X5 @ A2 )
           => ( ( member_set_nat2 @ Y3 @ A2 )
             => ( X5 = Y3 ) ) )
       => ( is_singleton_set_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_7992_is__singletonI_H,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
     => ( ! [X5: real,Y3: real] :
            ( ( member_real2 @ X5 @ A2 )
           => ( ( member_real2 @ Y3 @ A2 )
             => ( X5 = Y3 ) ) )
       => ( is_singleton_real @ A2 ) ) ) ).

% is_singletonI'
thf(fact_7993_is__singletonI_H,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ! [X5: $o,Y3: $o] :
            ( ( member_o2 @ X5 @ A2 )
           => ( ( member_o2 @ Y3 @ A2 )
             => ( X5 = Y3 ) ) )
       => ( is_singleton_o @ A2 ) ) ) ).

% is_singletonI'
thf(fact_7994_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X5: nat,Y3: nat] :
            ( ( member_nat2 @ X5 @ A2 )
           => ( ( member_nat2 @ Y3 @ A2 )
             => ( X5 = Y3 ) ) )
       => ( is_singleton_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_7995_is__singletonI_H,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
     => ( ! [X5: int,Y3: int] :
            ( ( member_int2 @ X5 @ A2 )
           => ( ( member_int2 @ Y3 @ A2 )
             => ( X5 = Y3 ) ) )
       => ( is_singleton_int @ A2 ) ) ) ).

% is_singletonI'
thf(fact_7996_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X5: nat] :
              ( ( ord_less_eq_nat @ C @ X5 )
             => ( ( ord_less_nat @ X5 @ D )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups3542108847815614940at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_7997_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: nat > real,H2: nat > real] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X5: nat] :
              ( ( ord_less_eq_nat @ C @ X5 )
             => ( ( ord_less_nat @ X5 @ D )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) ) )
         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups6591440286371151544t_real @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_7998_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: nat > int,H2: nat > int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X5: nat] :
              ( ( ord_less_eq_nat @ C @ X5 )
             => ( ( ord_less_nat @ X5 @ D )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) ) )
         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups705719431365010083at_int @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_7999_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D: nat,G: nat > nat,H2: nat > nat] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X5: nat] :
              ( ( ord_less_eq_nat @ C @ X5 )
             => ( ( ord_less_nat @ X5 @ D )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) ) )
         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups708209901874060359at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_8000_prod_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D: int,G: int > int,H2: int > int] :
      ( ( A = C )
     => ( ( B = D )
       => ( ! [X5: int] :
              ( ( ord_less_eq_int @ C @ X5 )
             => ( ( ord_less_int @ X5 @ D )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) ) )
         => ( ( groups1705073143266064639nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups1705073143266064639nt_int @ H2 @ ( set_or4662586982721622107an_int @ C @ D ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_8001_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8002_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8003_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8004_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8005_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P4: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) )
          = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_8006_sum__diff__nat__ivl,axiom,
    ! [M: nat,N2: nat,P4: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) )
          = ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_8007_size__list__estimation,axiom,
    ! [X: real,Xs: list_real,Y: nat,F: real > nat] :
      ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_real @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_8008_size__list__estimation,axiom,
    ! [X: $o,Xs: list_o,Y: nat,F: $o > nat] :
      ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_o @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_8009_size__list__estimation,axiom,
    ! [X: set_nat,Xs: list_set_nat,Y: nat,F: set_nat > nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_set_nat @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_8010_size__list__estimation,axiom,
    ! [X: int,Xs: list_int,Y: nat,F: int > nat] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_int @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_8011_size__list__estimation,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_8012_size__list__estimation,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: nat,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( ord_less_nat @ Y @ ( F @ X ) )
       => ( ord_less_nat @ Y @ ( size_list_VEBT_VEBT @ F @ Xs ) ) ) ) ).

% size_list_estimation
thf(fact_8013_size__list__pointwise,axiom,
    ! [Xs: list_real,F: real > nat,G: real > nat] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ ( set_real2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_nat @ ( size_list_real @ F @ Xs ) @ ( size_list_real @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_8014_size__list__pointwise,axiom,
    ! [Xs: list_o,F: $o > nat,G: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o2 @ X5 @ ( set_o2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_nat @ ( size_list_o @ F @ Xs ) @ ( size_list_o @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_8015_size__list__pointwise,axiom,
    ! [Xs: list_set_nat,F: set_nat > nat,G: set_nat > nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat2 @ X5 @ ( set_set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_nat @ ( size_list_set_nat @ F @ Xs ) @ ( size_list_set_nat @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_8016_size__list__pointwise,axiom,
    ! [Xs: list_int,F: int > nat,G: int > nat] :
      ( ! [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_nat @ ( size_list_int @ F @ Xs ) @ ( size_list_int @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_8017_size__list__pointwise,axiom,
    ! [Xs: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_8018_size__list__pointwise,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_nat @ ( size_list_VEBT_VEBT @ F @ Xs ) @ ( size_list_VEBT_VEBT @ G @ Xs ) ) ) ).

% size_list_pointwise
thf(fact_8019_size__list__estimation_H,axiom,
    ! [X: real,Xs: list_real,Y: nat,F: real > nat] :
      ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_real @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_8020_size__list__estimation_H,axiom,
    ! [X: $o,Xs: list_o,Y: nat,F: $o > nat] :
      ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_o @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_8021_size__list__estimation_H,axiom,
    ! [X: set_nat,Xs: list_set_nat,Y: nat,F: set_nat > nat] :
      ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_set_nat @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_8022_size__list__estimation_H,axiom,
    ! [X: int,Xs: list_int,Y: nat,F: int > nat] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_int @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_8023_size__list__estimation_H,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_8024_size__list__estimation_H,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: nat,F: vEBT_VEBT > nat] :
      ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
       => ( ord_less_eq_nat @ Y @ ( size_list_VEBT_VEBT @ F @ Xs ) ) ) ) ).

% size_list_estimation'
thf(fact_8025_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_8026_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_8027_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_8028_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_8029_prod_OatLeastLessThan__concat,axiom,
    ! [M: nat,N2: nat,P4: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_8030_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat2 @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_8031_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_8032_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le7203529160286727270d_enat @ ( set_or5403411693681687835d_enat @ A @ B ) @ ( set_or4374356025156299511d_enat @ C @ D ) )
      = ( ( ord_le2932123472753598470d_enat @ A @ B )
       => ( ( ord_le2932123472753598470d_enat @ C @ A )
          & ( ord_le72135733267957522d_enat @ B @ D ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8033_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: filter_nat,B: filter_nat,C: filter_nat,D: filter_nat] :
      ( ( ord_le2426478655948331894er_nat @ ( set_or1955772592623580779er_nat @ A @ B ) @ ( set_or1773934645810362255er_nat @ C @ D ) )
      = ( ( ord_le2510731241096832064er_nat @ A @ B )
       => ( ( ord_le2510731241096832064er_nat @ C @ A )
          & ( ord_less_filter_nat @ B @ D ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8034_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_set_nat @ B @ D ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8035_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D ) )
      = ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_real @ B @ D ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8036_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
      = ( ( ord_less_eq_nat @ A @ B )
       => ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_nat @ B @ D ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8037_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
      = ( ( ord_less_eq_int @ A @ B )
       => ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_int @ B @ D ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_8038_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ord_less_real @ A @ B )
       => ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_8039_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8040_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8041_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > extended_enat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z5237406670263579293d_enat )
     => ( ( groups7108830773950497114d_enat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7108830773950497114d_enat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8042_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8043_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8044_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8045_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8046_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8047_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8048_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8049_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ M ) @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8050_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8051_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8052_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8053_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8054_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8055_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8056_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or7139685690850216873Than_o
    = ( ^ [A3: $o,B3: $o] : ( minus_minus_set_o @ ( set_or8904488021354931149Most_o @ A3 @ B3 ) @ ( insert_o2 @ B3 @ bot_bot_set_o ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_8057_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or66887138388493659n_real
    = ( ^ [A3: real,B3: real] : ( minus_minus_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( insert_real2 @ B3 @ bot_bot_set_real ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_8058_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [A3: nat,B3: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( insert_nat2 @ B3 @ bot_bot_set_nat ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_8059_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [A3: int,B3: int] : ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( insert_int2 @ B3 @ bot_bot_set_int ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_8060_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_8061_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > complex,N2: nat] :
      ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_8062_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > extended_enat,N2: nat] :
      ( ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_8063_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_8064_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_8065_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_8066_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_complex @ ( G @ M ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_8067_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_7803423173614009249d_enat @ ( G @ M ) @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_8068_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_8069_prod_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_8070_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_8071_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_8072_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_8073_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_8074_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_8075_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ N2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_8076_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_p3455044024723400733d_enat @ ( G @ N2 ) @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_8077_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ N2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_8078_sum_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ N2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_8079_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( G @ N2 ) @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_8080_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( G @ N2 ) @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_8081_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > extended_enat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_7803423173614009249d_enat @ ( G @ N2 ) @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_8082_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( G @ N2 ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_8083_prod_Olast__plus,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_nat @ ( G @ N2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_8084_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( minus_minus_int @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8085_sum__Suc__diff_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( minus_minus_real @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8086_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I5 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_8087_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I5 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_8088_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = ( insert_nat2 @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_8089_sum_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( groups3542108847815614940at_nat @ ( A @ I5 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% sum.nested_swap
thf(fact_8090_sum_Onested__swap,axiom,
    ! [A: nat > nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( groups6591440286371151544t_real @ ( A @ I5 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% sum.nested_swap
thf(fact_8091_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I5 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_8092_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ ( suc @ I5 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_8093_prod_Onested__swap,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( groups705719431365010083at_int @ ( A @ I5 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_8094_prod_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( groups708209901874060359at_nat @ ( A @ I5 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I5: nat] : ( A @ I5 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_8095_sum_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [M2: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% sum.nat_group
thf(fact_8096_sum_Onat__group,axiom,
    ! [G: nat > real,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M2: nat] : ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% sum.nat_group
thf(fact_8097_prod_Onat__group,axiom,
    ! [G: nat > int,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [M2: nat] : ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% prod.nat_group
thf(fact_8098_prod_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [M2: nat] : ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M2 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% prod.nat_group
thf(fact_8099_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_Suc_fact
thf(fact_8100_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_8101_is__singletonE,axiom,
    ! [A2: set_real] :
      ( ( is_singleton_real @ A2 )
     => ~ ! [X5: real] :
            ( A2
           != ( insert_real2 @ X5 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_8102_is__singletonE,axiom,
    ! [A2: set_o] :
      ( ( is_singleton_o @ A2 )
     => ~ ! [X5: $o] :
            ( A2
           != ( insert_o2 @ X5 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_8103_is__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( is_singleton_nat @ A2 )
     => ~ ! [X5: nat] :
            ( A2
           != ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_8104_is__singletonE,axiom,
    ! [A2: set_int] :
      ( ( is_singleton_int @ A2 )
     => ~ ! [X5: int] :
            ( A2
           != ( insert_int2 @ X5 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_8105_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A4: set_real] :
        ? [X2: real] :
          ( A4
          = ( insert_real2 @ X2 @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_8106_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
        ? [X2: $o] :
          ( A4
          = ( insert_o2 @ X2 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_8107_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A4: set_nat] :
        ? [X2: nat] :
          ( A4
          = ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_8108_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A4: set_int] :
        ? [X2: int] :
          ( A4
          = ( insert_int2 @ X2 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_8109_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_8110_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_8111_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups7108830773950497114d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_p3455044024723400733d_enat @ ( groups7108830773950497114d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_8112_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_8113_sum_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_8114_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_8115_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_8116_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > extended_enat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_on7984719198319812577d_enat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups7961826882256487087d_enat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_7803423173614009249d_enat @ ( groups7961826882256487087d_enat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_8117_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_8118_prod_Ohead__if,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_8119_fact__prod__Suc,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_Suc
thf(fact_8120_fact__prod__Suc,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_Suc
thf(fact_8121_fact__prod__Suc,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N: nat] : ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_Suc
thf(fact_8122_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8123_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8124_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > int,N2: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8125_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8126_pochhammer__prod,axiom,
    ( comm_s3181272606743183617d_enat
    = ( ^ [A3: extended_enat,N: nat] :
          ( groups7961826882256487087d_enat
          @ ^ [I5: nat] : ( plus_p3455044024723400733d_enat @ A3 @ ( semiri4216267220026989637d_enat @ I5 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_8127_pochhammer__prod,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A3: real,N: nat] :
          ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ I5 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_8128_pochhammer__prod,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A3: int,N: nat] :
          ( groups705719431365010083at_int
          @ ^ [I5: nat] : ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ I5 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_8129_pochhammer__prod,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A3: nat,N: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I5: nat] : ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ I5 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% pochhammer_prod
thf(fact_8130_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_nat2 @ ( 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_8131_fact__prod__rev,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] : ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_rev
thf(fact_8132_fact__prod__rev,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] : ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_rev
thf(fact_8133_fact__prod__rev,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N: nat] : ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ ( minus_minus_nat @ N ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ).

% fact_prod_rev
thf(fact_8134_summable__Cauchy,axiom,
    ( summable_complex
    = ( ^ [F5: nat > complex] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [N5: nat] :
            ! [M2: nat] :
              ( ( ord_less_eq_nat @ N5 @ M2 )
             => ! [N: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F5 @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) @ E3 ) ) ) ) ) ).

% summable_Cauchy
thf(fact_8135_summable__Cauchy,axiom,
    ( summable_real
    = ( ^ [F5: nat > real] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [N5: nat] :
            ! [M2: nat] :
              ( ( ord_less_eq_nat @ N5 @ M2 )
             => ! [N: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F5 @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) @ E3 ) ) ) ) ) ).

% summable_Cauchy
thf(fact_8136_CauchyD,axiom,
    ! [X7: nat > complex,E2: real] :
      ( ( topolo6517432010174082258omplex @ X7 )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ? [M8: nat] :
          ! [M4: nat] :
            ( ( ord_less_eq_nat @ M8 @ M4 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M8 @ N6 )
               => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X7 @ M4 ) @ ( X7 @ N6 ) ) ) @ E2 ) ) ) ) ) ).

% CauchyD
thf(fact_8137_CauchyD,axiom,
    ! [X7: nat > real,E2: real] :
      ( ( topolo4055970368930404560y_real @ X7 )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ? [M8: nat] :
          ! [M4: nat] :
            ( ( ord_less_eq_nat @ M8 @ M4 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M8 @ N6 )
               => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X7 @ M4 ) @ ( X7 @ N6 ) ) ) @ E2 ) ) ) ) ) ).

% CauchyD
thf(fact_8138_CauchyI,axiom,
    ! [X7: nat > complex] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ? [M10: nat] :
            ! [M3: nat] :
              ( ( ord_less_eq_nat @ M10 @ M3 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N3 )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) ) @ E ) ) ) )
     => ( topolo6517432010174082258omplex @ X7 ) ) ).

% CauchyI
thf(fact_8139_CauchyI,axiom,
    ! [X7: nat > real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ? [M10: nat] :
            ! [M3: nat] :
              ( ( ord_less_eq_nat @ M10 @ M3 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N3 )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) ) @ E ) ) ) )
     => ( topolo4055970368930404560y_real @ X7 ) ) ).

% CauchyI
thf(fact_8140_Cauchy__iff,axiom,
    ( topolo6517432010174082258omplex
    = ( ^ [X8: nat > complex] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M9: nat] :
            ! [M2: nat] :
              ( ( ord_less_eq_nat @ M9 @ M2 )
             => ! [N: nat] :
                  ( ( ord_less_eq_nat @ M9 @ N )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M2 ) @ ( X8 @ N ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_8141_Cauchy__iff,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M9: nat] :
            ! [M2: nat] :
              ( ( ord_less_eq_nat @ M9 @ M2 )
             => ! [N: nat] :
                  ( ( ord_less_eq_nat @ M9 @ N )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M2 ) @ ( X8 @ N ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_8142_sums__group,axiom,
    ! [F: nat > nat,S: nat,K: nat] :
      ( ( sums_nat @ F @ S )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( sums_nat
          @ ^ [N: nat] : ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ N @ K ) @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ K ) ) )
          @ S ) ) ) ).

% sums_group
thf(fact_8143_sums__group,axiom,
    ! [F: nat > real,S: real,K: nat] :
      ( ( sums_real @ F @ S )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( sums_real
          @ ^ [N: nat] : ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ N @ K ) @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ K ) ) )
          @ S ) ) ) ).

% sums_group
thf(fact_8144_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_8145_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri5044797733671781792omplex @ N2 )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_8146_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri4449623510593786356d_enat @ N2 )
        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri4449623510593786356d_enat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_8147_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_8148_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_8149_fact__split,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri2265585572941072030t_real @ N2 )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ N2 @ K ) @ N2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_split
thf(fact_8150_binomial__altdef__of__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
        = ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I5 ) ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ K @ I5 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% binomial_altdef_of_nat
thf(fact_8151_gbinomial__altdef__of__nat,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K2: nat] :
          ( groups129246275422532515t_real
          @ ^ [I5: nat] : ( divide_divide_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ I5 ) ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ K2 @ I5 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% gbinomial_altdef_of_nat
thf(fact_8152_gbinomial__mult__fact,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( gbinomial_complex @ A @ K ) )
      = ( groups6464643781859351333omplex
        @ ^ [I5: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact
thf(fact_8153_gbinomial__mult__fact,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( gbinomial_real @ A @ K ) )
      = ( groups129246275422532515t_real
        @ ^ [I5: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact
thf(fact_8154_gbinomial__mult__fact_H,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( semiri5044797733671781792omplex @ K ) )
      = ( groups6464643781859351333omplex
        @ ^ [I5: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact'
thf(fact_8155_gbinomial__mult__fact_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( semiri2265585572941072030t_real @ K ) )
      = ( groups129246275422532515t_real
        @ ^ [I5: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I5 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ).

% gbinomial_mult_fact'
thf(fact_8156_gbinomial__prod__rev,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K2: nat] :
          ( divide_divide_real
          @ ( groups129246275422532515t_real
            @ ^ [I5: nat] : ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ I5 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
          @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8157_gbinomial__prod__rev,axiom,
    ( gbinomial_int
    = ( ^ [A3: int,K2: nat] :
          ( divide_divide_int
          @ ( groups705719431365010083at_int
            @ ^ [I5: nat] : ( minus_minus_int @ A3 @ ( semiri1314217659103216013at_int @ I5 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
          @ ( semiri1406184849735516958ct_int @ K2 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8158_gbinomial__prod__rev,axiom,
    ( gbinomial_nat
    = ( ^ [A3: nat,K2: nat] :
          ( divide_divide_nat
          @ ( groups708209901874060359at_nat
            @ ^ [I5: nat] : ( minus_minus_nat @ A3 @ ( semiri1316708129612266289at_nat @ I5 ) )
            @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
          @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ).

% gbinomial_prod_rev
thf(fact_8159_horner__sum__eq__sum,axiom,
    ( groups6842663049115397189BT_int
    = ( ^ [F5: vEBT_VEBT > int,A3: int,Xs2: list_VEBT_VEBT] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( times_times_int @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) @ ( power_power_int @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8160_horner__sum__eq__sum,axiom,
    ( groups8485231416243008693nt_int
    = ( ^ [F5: int > int,A3: int,Xs2: list_int] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( times_times_int @ ( F5 @ ( nth_int @ Xs2 @ N ) ) @ ( power_power_int @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8161_horner__sum__eq__sum,axiom,
    ( groups7485877704341954137at_int
    = ( ^ [F5: nat > int,A3: int,Xs2: list_nat] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( times_times_int @ ( F5 @ ( nth_nat @ Xs2 @ N ) ) @ ( power_power_int @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8162_horner__sum__eq__sum,axiom,
    ( groups1931381680841367751omplex
    = ( ^ [F5: vEBT_VEBT > complex,A3: complex,Xs2: list_VEBT_VEBT] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( times_times_complex @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) @ ( power_power_complex @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8163_horner__sum__eq__sum,axiom,
    ( groups1380173120649922871omplex
    = ( ^ [F5: int > complex,A3: complex,Xs2: list_int] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( times_times_complex @ ( F5 @ ( nth_int @ Xs2 @ N ) ) @ ( power_power_complex @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8164_horner__sum__eq__sum,axiom,
    ( groups404637655443745499omplex
    = ( ^ [F5: nat > complex,A3: complex,Xs2: list_nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( times_times_complex @ ( F5 @ ( nth_nat @ Xs2 @ N ) ) @ ( power_power_complex @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8165_horner__sum__eq__sum,axiom,
    ( groups3844460828123498381d_enat
    = ( ^ [F5: vEBT_VEBT > extended_enat,A3: extended_enat,Xs2: list_VEBT_VEBT] :
          ( groups7108830773950497114d_enat
          @ ^ [N: nat] : ( times_7803423173614009249d_enat @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) @ ( power_8040749407984259932d_enat @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8166_horner__sum__eq__sum,axiom,
    ( groups7888997386326813469d_enat
    = ( ^ [F5: int > extended_enat,A3: extended_enat,Xs2: list_int] :
          ( groups7108830773950497114d_enat
          @ ^ [N: nat] : ( times_7803423173614009249d_enat @ ( F5 @ ( nth_int @ Xs2 @ N ) ) @ ( power_8040749407984259932d_enat @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8167_horner__sum__eq__sum,axiom,
    ( groups1549203402269857401d_enat
    = ( ^ [F5: nat > extended_enat,A3: extended_enat,Xs2: list_nat] :
          ( groups7108830773950497114d_enat
          @ ^ [N: nat] : ( times_7803423173614009249d_enat @ ( F5 @ ( nth_nat @ Xs2 @ N ) ) @ ( power_8040749407984259932d_enat @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8168_horner__sum__eq__sum,axiom,
    ( groups6845153519624447465BT_nat
    = ( ^ [F5: vEBT_VEBT > nat,A3: nat,Xs2: list_VEBT_VEBT] :
          ( groups3542108847815614940at_nat
          @ ^ [N: nat] : ( times_times_nat @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) @ ( power_power_nat @ A3 @ N ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum
thf(fact_8169_Chebyshev__sum__upper,axiom,
    ! [N2: nat,A: nat > int,B: nat > int] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_int @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_int @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_int
          @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 )
            @ ( groups3539618377306564664at_int
              @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B @ K2 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_int @ ( groups3539618377306564664at_int @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3539618377306564664at_int @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper
thf(fact_8170_Chebyshev__sum__upper,axiom,
    ! [N2: nat,A: nat > real,B: nat > real] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_real @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_real @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_real
          @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 )
            @ ( groups6591440286371151544t_real
              @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B @ K2 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_real @ ( groups6591440286371151544t_real @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups6591440286371151544t_real @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper
thf(fact_8171_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I5: nat] : ( times_times_nat @ ( A @ I5 ) @ ( B @ I5 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_8172_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_8173_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N2: nat] :
      ( ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ ( set_list_VEBT_VEBT2 @ Xs ) )
         => ( ( size_s6755466524823107622T_VEBT @ X5 )
            = N2 ) )
     => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
        = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_8174_length__mul__elem,axiom,
    ! [Xs: list_list_int,N2: nat] :
      ( ! [X5: list_int] :
          ( ( member_list_int @ X5 @ ( set_list_int2 @ Xs ) )
         => ( ( size_size_list_int @ X5 )
            = N2 ) )
     => ( ( size_size_list_int @ ( concat_int @ Xs ) )
        = ( times_times_nat @ ( size_s533118279054570080st_int @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_8175_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N2: nat] :
      ( ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
         => ( ( size_size_list_nat @ X5 )
            = N2 ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_8176_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_8177_set__n__lists,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( ( set_list_VEBT_VEBT2 @ ( n_lists_VEBT_VEBT @ N2 @ Xs ) )
      = ( collec5608196760682091941T_VEBT
        @ ^ [Ys3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Ys3 )
              = N2 )
            & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Ys3 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_8178_set__n__lists,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( set_list_int2 @ ( n_lists_int @ N2 @ Xs ) )
      = ( collect_list_int
        @ ^ [Ys3: list_int] :
            ( ( ( size_size_list_int @ Ys3 )
              = N2 )
            & ( ord_less_eq_set_int @ ( set_int2 @ Ys3 ) @ ( set_int2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_8179_set__n__lists,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( set_list_nat2 @ ( n_lists_nat @ N2 @ Xs ) )
      = ( collect_list_nat
        @ ^ [Ys3: list_nat] :
            ( ( ( size_size_list_nat @ Ys3 )
              = N2 )
            & ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_8180_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_8181_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_8182_case__nat__add__eq__if,axiom,
    ! [A: $o,F: nat > $o,V: num,N2: nat] :
      ( ( case_nat_o @ A @ F @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_8183_case__nat__add__eq__if,axiom,
    ! [A: nat,F: nat > nat,V: num,N2: nat] :
      ( ( case_nat_nat @ A @ F @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_8184_nat_Ocase__distrib,axiom,
    ! [H2: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( H2 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( case_nat_o @ ( H2 @ F1 )
        @ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_8185_nat_Ocase__distrib,axiom,
    ! [H2: $o > nat,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( H2 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( case_nat_nat @ ( H2 @ F1 )
        @ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_8186_nat_Ocase__distrib,axiom,
    ! [H2: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( H2 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( case_nat_o @ ( H2 @ F1 )
        @ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_8187_nat_Ocase__distrib,axiom,
    ! [H2: nat > nat,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( H2 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( case_nat_nat @ ( H2 @ F1 )
        @ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
        @ Nat ) ) ).

% nat.case_distrib
thf(fact_8188_old_Onat_Osimps_I5_J,axiom,
    ! [F1: $o,F22: nat > $o,X22: nat] :
      ( ( case_nat_o @ F1 @ F22 @ ( suc @ X22 ) )
      = ( F22 @ X22 ) ) ).

% old.nat.simps(5)
thf(fact_8189_old_Onat_Osimps_I5_J,axiom,
    ! [F1: nat,F22: nat > nat,X22: nat] :
      ( ( case_nat_nat @ F1 @ F22 @ ( suc @ X22 ) )
      = ( F22 @ X22 ) ) ).

% old.nat.simps(5)
thf(fact_8190_old_Onat_Osimps_I4_J,axiom,
    ! [F1: $o,F22: nat > $o] :
      ( ( case_nat_o @ F1 @ F22 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_8191_old_Onat_Osimps_I4_J,axiom,
    ! [F1: nat,F22: nat > nat] :
      ( ( case_nat_nat @ F1 @ F22 @ zero_zero_nat )
      = F1 ) ).

% old.nat.simps(4)
thf(fact_8192_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_8193_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_8194_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_8195_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N2 @ M5 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_8196_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_8197_length__n__lists__elem,axiom,
    ! [Ys: list_VEBT_VEBT,N2: nat,Xs: list_VEBT_VEBT] :
      ( ( member2936631157270082147T_VEBT @ Ys @ ( set_list_VEBT_VEBT2 @ ( n_lists_VEBT_VEBT @ N2 @ Xs ) ) )
     => ( ( size_s6755466524823107622T_VEBT @ Ys )
        = N2 ) ) ).

% length_n_lists_elem
thf(fact_8198_length__n__lists__elem,axiom,
    ! [Ys: list_int,N2: nat,Xs: list_int] :
      ( ( member_list_int @ Ys @ ( set_list_int2 @ ( n_lists_int @ N2 @ Xs ) ) )
     => ( ( size_size_list_int @ Ys )
        = N2 ) ) ).

% length_n_lists_elem
thf(fact_8199_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N2: nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N2 @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N2 ) ) ).

% length_n_lists_elem
thf(fact_8200_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K2: nat] : K2
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_8201_Nitpick_Ocase__nat__unfold,axiom,
    ( case_nat_o
    = ( ^ [X2: $o,F5: nat > $o,N: nat] :
          ( ( ( N = zero_zero_nat )
           => X2 )
          & ( ( N != zero_zero_nat )
           => ( F5 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% Nitpick.case_nat_unfold
thf(fact_8202_Nitpick_Ocase__nat__unfold,axiom,
    ( case_nat_nat
    = ( ^ [X2: nat,F5: nat > nat,N: nat] : ( if_nat @ ( N = zero_zero_nat ) @ X2 @ ( F5 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% Nitpick.case_nat_unfold
thf(fact_8203_nat_Osplit__sels_I1_J,axiom,
    ! [P2: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( P2 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( ( ( Nat = zero_zero_nat )
         => ( P2 @ F1 ) )
        & ( ( Nat
            = ( suc @ ( pred @ Nat ) ) )
         => ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).

% nat.split_sels(1)
thf(fact_8204_nat_Osplit__sels_I1_J,axiom,
    ! [P2: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( P2 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( ( ( Nat = zero_zero_nat )
         => ( P2 @ F1 ) )
        & ( ( Nat
            = ( suc @ ( pred @ Nat ) ) )
         => ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).

% nat.split_sels(1)
thf(fact_8205_nat_Osplit__sels_I2_J,axiom,
    ! [P2: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
      ( ( P2 @ ( case_nat_o @ F1 @ F22 @ Nat ) )
      = ( ~ ( ( ( Nat = zero_zero_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Nat
                = ( suc @ ( pred @ Nat ) ) )
              & ~ ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).

% nat.split_sels(2)
thf(fact_8206_nat_Osplit__sels_I2_J,axiom,
    ! [P2: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
      ( ( P2 @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
      = ( ~ ( ( ( Nat = zero_zero_nat )
              & ~ ( P2 @ F1 ) )
            | ( ( Nat
                = ( suc @ ( pred @ Nat ) ) )
              & ~ ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).

% nat.split_sels(2)
thf(fact_8207_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_8208_subset__CollectI,axiom,
    ! [B2: set_o,A2: set_o,Q: $o > $o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ B2 @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o2 @ X5 @ B2 )
           => ( ( Q @ X5 )
             => ( P2 @ X5 ) ) )
       => ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ B2 )
                & ( Q @ X2 ) ) )
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_8209_subset__CollectI,axiom,
    ! [B2: set_real,A2: set_real,Q: real > $o,P2: real > $o] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( ! [X5: real] :
            ( ( member_real2 @ X5 @ B2 )
           => ( ( Q @ X5 )
             => ( P2 @ X5 ) ) )
       => ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ B2 )
                & ( Q @ X2 ) ) )
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_8210_subset__CollectI,axiom,
    ! [B2: set_list_nat,A2: set_list_nat,Q: list_nat > $o,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
     => ( ! [X5: list_nat] :
            ( ( member_list_nat @ X5 @ B2 )
           => ( ( Q @ X5 )
             => ( P2 @ X5 ) ) )
       => ( ord_le6045566169113846134st_nat
          @ ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( member_list_nat @ X2 @ B2 )
                & ( Q @ X2 ) ) )
          @ ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( member_list_nat @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_8211_subset__CollectI,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,Q: set_nat > $o,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ! [X5: set_nat] :
            ( ( member_set_nat2 @ X5 @ B2 )
           => ( ( Q @ X5 )
             => ( P2 @ X5 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( member_set_nat2 @ X2 @ B2 )
                & ( Q @ X2 ) ) )
          @ ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( member_set_nat2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_8212_subset__CollectI,axiom,
    ! [B2: set_int,A2: set_int,Q: int > $o,P2: int > $o] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ! [X5: int] :
            ( ( member_int2 @ X5 @ B2 )
           => ( ( Q @ X5 )
             => ( P2 @ X5 ) ) )
       => ( ord_less_eq_set_int
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ B2 )
                & ( Q @ X2 ) ) )
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_8213_subset__CollectI,axiom,
    ! [B2: set_nat,A2: set_nat,Q: nat > $o,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat2 @ X5 @ B2 )
           => ( ( Q @ X5 )
             => ( P2 @ X5 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat2 @ X2 @ B2 )
                & ( Q @ X2 ) ) )
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_8214_subset__Collect__iff,axiom,
    ! [B2: set_o,A2: set_o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ B2 @ A2 )
     => ( ( ord_less_eq_set_o @ B2
          @ ( collect_o
            @ ^ [X2: $o] :
                ( ( member_o2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: $o] :
              ( ( member_o2 @ X2 @ B2 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_8215_subset__Collect__iff,axiom,
    ! [B2: set_real,A2: set_real,P2: real > $o] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( ( ord_less_eq_set_real @ B2
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: real] :
              ( ( member_real2 @ X2 @ B2 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_8216_subset__Collect__iff,axiom,
    ! [B2: set_list_nat,A2: set_list_nat,P2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
     => ( ( ord_le6045566169113846134st_nat @ B2
          @ ( collect_list_nat
            @ ^ [X2: list_nat] :
                ( ( member_list_nat @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: list_nat] :
              ( ( member_list_nat @ X2 @ B2 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_8217_subset__Collect__iff,axiom,
    ! [B2: set_set_nat,A2: set_set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ B2
          @ ( collect_set_nat
            @ ^ [X2: set_nat] :
                ( ( member_set_nat2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: set_nat] :
              ( ( member_set_nat2 @ X2 @ B2 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_8218_subset__Collect__iff,axiom,
    ! [B2: set_int,A2: set_int,P2: int > $o] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( ord_less_eq_set_int @ B2
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: int] :
              ( ( member_int2 @ X2 @ B2 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_8219_subset__Collect__iff,axiom,
    ! [B2: set_nat,A2: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ B2
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat2 @ X2 @ A2 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: nat] :
              ( ( member_nat2 @ X2 @ B2 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_8220_signed__take__bit__code,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 ) @ N ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 ) @ ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8221_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_8222_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K2 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K2 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M2 @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_8223_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_8224_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_8225_push__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% push_bit_of_0
thf(fact_8226_push__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% push_bit_of_0
thf(fact_8227_push__bit__eq__0__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( bit_se547839408752420682it_nat @ N2 @ A )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
thf(fact_8228_push__bit__eq__0__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( ( bit_se545348938243370406it_int @ N2 @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
thf(fact_8229_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_8230_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_8231_push__bit__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( bit_se547839408752420682it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_8232_push__bit__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ K ) )
      = ( bit_se545348938243370406it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_8233_push__bit__Suc__minus__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_Suc_minus_numeral
thf(fact_8234_push__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N2 ) @ A )
      = ( bit_se547839408752420682it_nat @ N2 @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_8235_push__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ A )
      = ( bit_se545348938243370406it_int @ N2 @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_8236_push__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_Suc_0
thf(fact_8237_even__push__bit__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( ( N2 != zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_8238_even__push__bit__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( ( N2 != zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_8239_push__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_8240_push__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_8241_fst__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] :
      ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_8242_fst__conv,axiom,
    ! [X1: vEBT_VEBT,X22: nat] :
      ( ( produc8713918199166443969BT_nat @ ( produc738532404422230701BT_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_8243_fst__conv,axiom,
    ! [X1: vEBT_VEBT,X22: extended_enat] :
      ( ( produc967593531271825845d_enat @ ( produc581526299967858633d_enat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_8244_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_8245_fst__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_8246_fst__eqD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_8247_fst__eqD,axiom,
    ! [X: vEBT_VEBT,Y: nat,A: vEBT_VEBT] :
      ( ( ( produc8713918199166443969BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_8248_fst__eqD,axiom,
    ! [X: vEBT_VEBT,Y: extended_enat,A: vEBT_VEBT] :
      ( ( ( produc967593531271825845d_enat @ ( produc581526299967858633d_enat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_8249_fst__eqD,axiom,
    ! [X: nat,Y: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_8250_fst__eqD,axiom,
    ! [X: int,Y: int,A: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_8251_push__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_8252_push__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_8253_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_8254_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_8255_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A3: nat,N: nat] :
          ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
         != zero_zero_nat ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_8256_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A3: int,N: nat] :
          ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
         != zero_zero_int ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_8257_take__bit__sum,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A3: nat] :
          ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( bit_se547839408752420682it_nat @ K2 @ ( zero_n2687167440665602831ol_nat @ ( bit_se1148574629649215175it_nat @ A3 @ K2 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% take_bit_sum
thf(fact_8258_take__bit__sum,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A3: int] :
          ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( bit_se545348938243370406it_int @ K2 @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ A3 @ K2 ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% take_bit_sum
thf(fact_8259_in__set__enumerate__eq,axiom,
    ! [P4: produc8025551001238799321T_VEBT,N2: nat,Xs: list_VEBT_VEBT] :
      ( ( member8549952807677709168T_VEBT @ P4 @ ( set_Pr5984661752051438084T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( produc8575180428842422559T_VEBT @ P4 ) )
        & ( ord_less_nat @ ( produc8575180428842422559T_VEBT @ P4 ) @ ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N2 ) )
        & ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ ( produc8575180428842422559T_VEBT @ P4 ) @ N2 ) )
          = ( produc8172668247895388509T_VEBT @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_8260_in__set__enumerate__eq,axiom,
    ! [P4: product_prod_nat_int,N2: nat,Xs: list_int] :
      ( ( member4262671552274231302at_int @ P4 @ ( set_Pr1470767568048878706at_int @ ( enumerate_int @ N2 @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_int @ P4 ) )
        & ( ord_less_nat @ ( product_fst_nat_int @ P4 ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N2 ) )
        & ( ( nth_int @ Xs @ ( minus_minus_nat @ ( product_fst_nat_int @ P4 ) @ N2 ) )
          = ( product_snd_nat_int @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_8261_in__set__enumerate__eq,axiom,
    ! [P4: product_prod_nat_nat,N2: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ P4 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N2 @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_nat @ P4 ) )
        & ( ord_less_nat @ ( product_fst_nat_nat @ P4 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N2 ) )
        & ( ( nth_nat @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ P4 ) @ N2 ) )
          = ( product_snd_nat_nat @ P4 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_8262_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_8263_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N2 )
        = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_8264_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rotate1_int @ Xs ) @ N2 )
        = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_8265_nth__rotate1,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).

% nth_rotate1
thf(fact_8266_set__update__distinct,axiom,
    ! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X ) )
          = ( insert_VEBT_VEBT2 @ X @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT2 @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_8267_set__update__distinct,axiom,
    ! [Xs: list_real,N2: nat,X: real] :
      ( ( distinct_real @ Xs )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
       => ( ( set_real2 @ ( list_update_real @ Xs @ N2 @ X ) )
          = ( insert_real2 @ X @ ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real2 @ ( nth_real @ Xs @ N2 ) @ bot_bot_set_real ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_8268_set__update__distinct,axiom,
    ! [Xs: list_o,N2: nat,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ Xs @ N2 @ X ) )
          = ( insert_o2 @ X @ ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ ( nth_o @ Xs @ N2 ) @ bot_bot_set_o ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_8269_set__update__distinct,axiom,
    ! [Xs: list_int,N2: nat,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ Xs @ N2 @ X ) )
          = ( insert_int2 @ X @ ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ ( nth_int @ Xs @ N2 ) @ bot_bot_set_int ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_8270_set__update__distinct,axiom,
    ! [Xs: list_nat,N2: nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) )
          = ( insert_nat2 @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ N2 ) @ bot_bot_set_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_8271_drop__bit__rec,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N: nat,A3: nat] : ( if_nat @ ( N = zero_zero_nat ) @ A3 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_8272_drop__bit__rec,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ A3 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_8273_drop__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% drop_bit_of_0
thf(fact_8274_drop__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_of_0
thf(fact_8275_drop__bit__drop__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se8570568707652914677it_nat @ M @ ( bit_se8570568707652914677it_nat @ N2 @ A ) )
      = ( bit_se8570568707652914677it_nat @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% drop_bit_drop_bit
thf(fact_8276_drop__bit__drop__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se8568078237143864401it_int @ N2 @ A ) )
      = ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% drop_bit_drop_bit
thf(fact_8277_set__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( rotate1_VEBT_VEBT @ Xs ) )
      = ( set_VEBT_VEBT2 @ Xs ) ) ).

% set_rotate1
thf(fact_8278_set__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( set_int2 @ ( rotate1_int @ Xs ) )
      = ( set_int2 @ Xs ) ) ).

% set_rotate1
thf(fact_8279_set__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2 @ ( rotate1_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_rotate1
thf(fact_8280_length__rotate1,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_rotate1
thf(fact_8281_length__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( rotate1_int @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_rotate1
thf(fact_8282_length__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rotate1
thf(fact_8283_distinct1__rotate,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ ( rotate1_int @ Xs ) )
      = ( distinct_int @ Xs ) ) ).

% distinct1_rotate
thf(fact_8284_distinct1__rotate,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ ( rotate1_nat @ Xs ) )
      = ( distinct_nat @ Xs ) ) ).

% distinct1_rotate
thf(fact_8285_distinct__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( distinct_int @ ( insert_int @ X @ Xs ) )
      = ( distinct_int @ Xs ) ) ).

% distinct_insert
thf(fact_8286_distinct__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( insert_nat @ X @ Xs ) )
      = ( distinct_nat @ Xs ) ) ).

% distinct_insert
thf(fact_8287_prod_Ocollapse,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8288_prod_Ocollapse,axiom,
    ! [Prod: produc9072475918466114483BT_nat] :
      ( ( produc738532404422230701BT_nat @ ( produc8713918199166443969BT_nat @ Prod ) @ ( produc8311406018219409919BT_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8289_prod_Ocollapse,axiom,
    ! [Prod: produc7272778201969148633d_enat] :
      ( ( produc581526299967858633d_enat @ ( produc967593531271825845d_enat @ Prod ) @ ( produc522021263368680183d_enat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8290_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8291_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] :
      ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8292_drop__bit__of__bool,axiom,
    ! [N2: nat,B: $o] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ B ) )
      = ( zero_n2687167440665602831ol_nat
        @ ( ( N2 = zero_zero_nat )
          & B ) ) ) ).

% drop_bit_of_bool
thf(fact_8293_drop__bit__of__bool,axiom,
    ! [N2: nat,B: $o] :
      ( ( bit_se8568078237143864401it_int @ N2 @ ( zero_n2684676970156552555ol_int @ B ) )
      = ( zero_n2684676970156552555ol_int
        @ ( ( N2 = zero_zero_nat )
          & B ) ) ) ).

% drop_bit_of_bool
thf(fact_8294_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_8295_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_8296_drop__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8570568707652914677it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( bit_se8570568707652914677it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_8297_drop__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( numeral_numeral_int @ K ) ) ) ).

% drop_bit_Suc_bit0
thf(fact_8298_drop__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8570568707652914677it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( bit_se8570568707652914677it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_8299_drop__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( numeral_numeral_int @ K ) ) ) ).

% drop_bit_Suc_bit1
thf(fact_8300_drop__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_8301_drop__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_8302_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_8303_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate1_int @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_8304_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_8305_distinct__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 ) )
       => ( ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( distinct_VEBT_VEBT @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_8306_distinct__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 ) )
       => ( ( distinct_int @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( distinct_int @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_8307_distinct__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 ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_8308_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_8309_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ( size_s3451745648224563538omplex @ Xs2 )
                = N2 )
              & ( distinct_complex @ Xs2 )
              & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8310_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( finite1862508098717546133d_enat
        @ ( collec8433460942617342167d_enat
          @ ^ [Xs2: list_Extended_enat] :
              ( ( ( size_s3941691890525107288d_enat @ Xs2 )
                = N2 )
              & ( distin4523846830085650399d_enat @ Xs2 )
              & ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8311_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N2 )
              & ( distinct_VEBT_VEBT @ Xs2 )
              & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8312_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ( size_size_list_int @ Xs2 )
                = N2 )
              & ( distinct_int @ Xs2 )
              & ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8313_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = N2 )
              & ( distinct_nat @ Xs2 )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_8314_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ Xs )
     => ( distinct_int @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_8315_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_8316_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( distinct_int @ Xs )
     => ( ( distinct_int @ Ys )
       => ( distin3744728255968310194nt_int @ ( product_int_int @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_8317_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( distinct_int @ Xs )
     => ( ( distinct_nat @ Ys )
       => ( distin7922579275477506902nt_nat @ ( product_int_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_8318_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( distinct_nat @ Xs )
     => ( ( distinct_int @ Ys )
       => ( distin2745374544067255638at_int @ ( product_nat_int @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_8319_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( distinct_nat @ Ys )
       => ( distin6923225563576452346at_nat @ ( product_nat_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_8320_snd__eqD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_8321_snd__eqD,axiom,
    ! [X: vEBT_VEBT,Y: nat,A: nat] :
      ( ( ( produc8311406018219409919BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_8322_snd__eqD,axiom,
    ! [X: vEBT_VEBT,Y: extended_enat,A: extended_enat] :
      ( ( ( produc522021263368680183d_enat @ ( produc581526299967858633d_enat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_8323_snd__eqD,axiom,
    ! [X: nat,Y: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_8324_snd__eqD,axiom,
    ! [X: int,Y: int,A: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_8325_snd__conv,axiom,
    ! [X1: product_prod_nat_nat,X22: product_prod_nat_nat] :
      ( ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_8326_snd__conv,axiom,
    ! [X1: vEBT_VEBT,X22: nat] :
      ( ( produc8311406018219409919BT_nat @ ( produc738532404422230701BT_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_8327_snd__conv,axiom,
    ! [X1: vEBT_VEBT,X22: extended_enat] :
      ( ( produc522021263368680183d_enat @ ( produc581526299967858633d_enat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_8328_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_8329_snd__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_8330_finite__distinct__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( ( set_VEBT_VEBT2 @ Xs3 )
            = A2 )
          & ( distinct_VEBT_VEBT @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8331_finite__distinct__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( ( set_nat2 @ Xs3 )
            = A2 )
          & ( distinct_nat @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8332_finite__distinct__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( ( set_complex2 @ Xs3 )
            = A2 )
          & ( distinct_complex @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8333_finite__distinct__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( ( set_int2 @ Xs3 )
            = A2 )
          & ( distinct_int @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8334_finite__distinct__list,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ? [Xs3: list_Extended_enat] :
          ( ( ( set_Extended_enat2 @ Xs3 )
            = A2 )
          & ( distin4523846830085650399d_enat @ Xs3 ) ) ) ).

% finite_distinct_list
thf(fact_8335_prod_Oexhaust__sel,axiom,
    ! [Prod: produc859450856879609959at_nat] :
      ( Prod
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ Prod ) @ ( produc6408287024330202629at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_8336_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9072475918466114483BT_nat] :
      ( Prod
      = ( produc738532404422230701BT_nat @ ( produc8713918199166443969BT_nat @ Prod ) @ ( produc8311406018219409919BT_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_8337_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7272778201969148633d_enat] :
      ( Prod
      = ( produc581526299967858633d_enat @ ( produc967593531271825845d_enat @ Prod ) @ ( produc522021263368680183d_enat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_8338_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_8339_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] :
      ( Prod
      = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_8340_surjective__pairing,axiom,
    ! [T: produc859450856879609959at_nat] :
      ( T
      = ( produc6161850002892822231at_nat @ ( produc3213797794245857475at_nat @ T ) @ ( produc6408287024330202629at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_8341_surjective__pairing,axiom,
    ! [T: produc9072475918466114483BT_nat] :
      ( T
      = ( produc738532404422230701BT_nat @ ( produc8713918199166443969BT_nat @ T ) @ ( produc8311406018219409919BT_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_8342_surjective__pairing,axiom,
    ! [T: produc7272778201969148633d_enat] :
      ( T
      = ( produc581526299967858633d_enat @ ( produc967593531271825845d_enat @ T ) @ ( produc522021263368680183d_enat @ T ) ) ) ).

% surjective_pairing
thf(fact_8343_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] :
      ( T
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T ) @ ( product_snd_nat_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_8344_surjective__pairing,axiom,
    ! [T: product_prod_int_int] :
      ( T
      = ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).

% surjective_pairing
thf(fact_8345_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,Y: product_prod_nat_nat,A: produc859450856879609959at_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc6161850002892822231at_nat @ X @ Y ) )
       => ( P2 @ ( produc3213797794245857475at_nat @ A ) @ ( produc6408287024330202629at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_8346_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: vEBT_VEBT > nat > $o,X: vEBT_VEBT,Y: nat,A: produc9072475918466114483BT_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc738532404422230701BT_nat @ X @ Y ) )
       => ( P2 @ ( produc8713918199166443969BT_nat @ A ) @ ( produc8311406018219409919BT_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_8347_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: vEBT_VEBT > extended_enat > $o,X: vEBT_VEBT,Y: extended_enat,A: produc7272778201969148633d_enat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc581526299967858633d_enat @ X @ Y ) )
       => ( P2 @ ( produc967593531271825845d_enat @ A ) @ ( produc522021263368680183d_enat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_8348_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( product_Pair_nat_nat @ X @ Y ) )
       => ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_8349_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: int > int > $o,X: int,Y: int,A: product_prod_int_int] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( product_Pair_int_int @ X @ Y ) )
       => ( P2 @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_8350_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = A )
      = ( ( bit_se8570568707652914677it_nat @ N2 @ A )
        = zero_zero_nat ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_8351_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [N2: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = A )
      = ( ( bit_se8568078237143864401it_int @ N2 @ A )
        = zero_zero_int ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_8352_take__bit__drop__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se8570568707652914677it_nat @ N2 @ A ) )
      = ( bit_se8570568707652914677it_nat @ N2 @ ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ) ).

% take_bit_drop_bit
thf(fact_8353_take__bit__drop__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_se8568078237143864401it_int @ N2 @ A ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ) ).

% take_bit_drop_bit
thf(fact_8354_subseqs__distinctD,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( member_list_int @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) )
     => ( ( distinct_int @ Xs )
       => ( distinct_int @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_8355_subseqs__distinctD,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( ( distinct_nat @ Xs )
       => ( distinct_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_8356_divides__aux__def,axiom,
    ( unique6322359934112328802ux_nat
    = ( ^ [Qr: product_prod_nat_nat] :
          ( ( product_snd_nat_nat @ Qr )
          = zero_zero_nat ) ) ) ).

% divides_aux_def
thf(fact_8357_divides__aux__def,axiom,
    ( unique6319869463603278526ux_int
    = ( ^ [Qr: product_prod_int_int] :
          ( ( product_snd_int_int @ Qr )
          = zero_zero_int ) ) ) ).

% divides_aux_def
thf(fact_8358_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBT
    = ( ^ [Xs2: list_VEBT_VEBT] :
        ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( I5 != J3 )
               => ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                 != ( nth_VEBT_VEBT @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_8359_distinct__conv__nth,axiom,
    ( distinct_int
    = ( ^ [Xs2: list_int] :
        ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs2 ) )
             => ( ( I5 != J3 )
               => ( ( nth_int @ Xs2 @ I5 )
                 != ( nth_int @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_8360_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
         => ! [J3: nat] :
              ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I5 != J3 )
               => ( ( nth_nat @ Xs2 @ I5 )
                 != ( nth_nat @ Xs2 @ J3 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_8361_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,J: nat] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( ( ( nth_VEBT_VEBT @ Xs @ I )
              = ( nth_VEBT_VEBT @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_8362_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( distinct_int @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ( ( nth_int @ Xs @ I )
              = ( nth_int @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_8363_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_8364_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_8365_prod_Osplit__sel__asm,axiom,
    ! [P2: nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_8366_prod_Osplit__sel__asm,axiom,
    ! [P2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_8367_prod_Osplit__sel__asm,axiom,
    ! [P2: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_8368_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( P2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_8369_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_8370_prod_Osplit__sel,axiom,
    ! [P2: nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_8371_prod_Osplit__sel,axiom,
    ! [P2: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_8372_prod_Osplit__sel,axiom,
    ! [P2: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_8373_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( P2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_8374_bits__ident,axiom,
    ! [N2: nat,A: nat] :
      ( ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ ( bit_se8570568707652914677it_nat @ N2 @ A ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = A ) ).

% bits_ident
thf(fact_8375_bits__ident,axiom,
    ! [N2: nat,A: int] :
      ( ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ ( bit_se8568078237143864401it_int @ N2 @ A ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = A ) ).

% bits_ident
thf(fact_8376_distinct__Ex1,axiom,
    ! [Xs: list_real,X: real] :
      ( ( distinct_real @ Xs )
     => ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ? [X5: nat] :
            ( ( ord_less_nat @ X5 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ X5 )
              = X )
            & ! [Y4: nat] :
                ( ( ( ord_less_nat @ Y4 @ ( size_size_list_real @ Xs ) )
                  & ( ( nth_real @ Xs @ Y4 )
                    = X ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_8377_distinct__Ex1,axiom,
    ! [Xs: list_o,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ? [X5: nat] :
            ( ( ord_less_nat @ X5 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ X5 )
              = X )
            & ! [Y4: nat] :
                ( ( ( ord_less_nat @ Y4 @ ( size_size_list_o @ Xs ) )
                  & ( ( nth_o @ Xs @ Y4 )
                    = X ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_8378_distinct__Ex1,axiom,
    ! [Xs: list_set_nat,X: set_nat] :
      ( ( distinct_set_nat @ Xs )
     => ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
       => ? [X5: nat] :
            ( ( ord_less_nat @ X5 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ X5 )
              = X )
            & ! [Y4: nat] :
                ( ( ( ord_less_nat @ Y4 @ ( size_s3254054031482475050et_nat @ Xs ) )
                  & ( ( nth_set_nat @ Xs @ Y4 )
                    = X ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_8379_distinct__Ex1,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ? [X5: nat] :
            ( ( ord_less_nat @ X5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ X5 )
              = X )
            & ! [Y4: nat] :
                ( ( ( ord_less_nat @ Y4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                  & ( ( nth_VEBT_VEBT @ Xs @ Y4 )
                    = X ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_8380_distinct__Ex1,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ? [X5: nat] :
            ( ( ord_less_nat @ X5 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ X5 )
              = X )
            & ! [Y4: nat] :
                ( ( ( ord_less_nat @ Y4 @ ( size_size_list_int @ Xs ) )
                  & ( ( nth_int @ Xs @ Y4 )
                    = X ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_8381_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ? [X5: nat] :
            ( ( ord_less_nat @ X5 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X5 )
              = X )
            & ! [Y4: nat] :
                ( ( ( ord_less_nat @ Y4 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y4 )
                    = X ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_8382_update__zip,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,I: nat,Xy: produc8243902056947475879T_VEBT] :
      ( ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I @ Xy )
      = ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( produc6118266651365575811T_VEBT @ Xy ) ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I @ ( produc351232539757196229T_VEBT @ Xy ) ) ) ) ).

% update_zip
thf(fact_8383_update__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat,I: nat,Xy: product_prod_nat_nat] :
      ( ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I @ Xy )
      = ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ ( product_fst_nat_nat @ Xy ) ) @ ( list_update_nat @ Ys @ I @ ( product_snd_nat_nat @ Xy ) ) ) ) ).

% update_zip
thf(fact_8384_update__zip,axiom,
    ! [Xs: list_int,Ys: list_int,I: nat,Xy: product_prod_int_int] :
      ( ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ Xy )
      = ( zip_int_int @ ( list_update_int @ Xs @ I @ ( product_fst_int_int @ Xy ) ) @ ( list_update_int @ Ys @ I @ ( product_snd_int_int @ Xy ) ) ) ) ).

% update_zip
thf(fact_8385_drop__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se8570568707652914677it_nat @ ( suc @ N2 ) @ A )
      = ( bit_se8570568707652914677it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_8386_drop__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ A )
      = ( bit_se8568078237143864401it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% drop_bit_Suc
thf(fact_8387_distinct__list__update,axiom,
    ! [Xs: list_set_nat,A: set_nat,I: nat] :
      ( ( distinct_set_nat @ Xs )
     => ( ~ ( member_set_nat2 @ A @ ( minus_2163939370556025621et_nat @ ( set_set_nat2 @ Xs ) @ ( insert_set_nat2 @ ( nth_set_nat @ Xs @ I ) @ bot_bot_set_set_nat ) ) )
       => ( distinct_set_nat @ ( list_update_set_nat @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_8388_distinct__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,A: vEBT_VEBT,I: nat] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ~ ( member_VEBT_VEBT2 @ A @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ bot_bo8194388402131092736T_VEBT ) ) )
       => ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_8389_distinct__list__update,axiom,
    ! [Xs: list_real,A: real,I: nat] :
      ( ( distinct_real @ Xs )
     => ( ~ ( member_real2 @ A @ ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real2 @ ( nth_real @ Xs @ I ) @ bot_bot_set_real ) ) )
       => ( distinct_real @ ( list_update_real @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_8390_distinct__list__update,axiom,
    ! [Xs: list_o,A: $o,I: nat] :
      ( ( distinct_o @ Xs )
     => ( ~ ( member_o2 @ A @ ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ ( nth_o @ Xs @ I ) @ bot_bot_set_o ) ) )
       => ( distinct_o @ ( list_update_o @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_8391_distinct__list__update,axiom,
    ! [Xs: list_int,A: int,I: nat] :
      ( ( distinct_int @ Xs )
     => ( ~ ( member_int2 @ A @ ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ ( nth_int @ Xs @ I ) @ bot_bot_set_int ) ) )
       => ( distinct_int @ ( list_update_int @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_8392_distinct__list__update,axiom,
    ! [Xs: list_nat,A: nat,I: nat] :
      ( ( distinct_nat @ Xs )
     => ( ~ ( member_nat2 @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ ( nth_nat @ Xs @ I ) @ bot_bot_set_nat ) ) )
       => ( distinct_nat @ ( list_update_nat @ Xs @ I @ A ) ) ) ) ).

% distinct_list_update
thf(fact_8393_in__set__zip,axiom,
    ! [P4: produc8243902056947475879T_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ P4 @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_VEBT_VEBT @ Xs @ N )
              = ( produc6118266651365575811T_VEBT @ P4 ) )
            & ( ( nth_VEBT_VEBT @ Ys @ N )
              = ( produc351232539757196229T_VEBT @ P4 ) )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8394_in__set__zip,axiom,
    ! [P4: produc4894624898956917775BT_int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( member5419026705395827622BT_int @ P4 @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_VEBT_VEBT @ Xs @ N )
              = ( produc8711427728657393693BT_int @ P4 ) )
            & ( ( nth_int @ Ys @ N )
              = ( produc8308915547710359643BT_int @ P4 ) )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8395_in__set__zip,axiom,
    ! [P4: produc9072475918466114483BT_nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( member373505688050248522BT_nat @ P4 @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_VEBT_VEBT @ Xs @ N )
              = ( produc8713918199166443969BT_nat @ P4 ) )
            & ( ( nth_nat @ Ys @ N )
              = ( produc8311406018219409919BT_nat @ P4 ) )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8396_in__set__zip,axiom,
    ! [P4: produc1531783533982839933T_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( member2056185340421749780T_VEBT @ P4 @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_int @ Xs @ N )
              = ( produc2081412961586463171T_VEBT @ P4 ) )
            & ( ( nth_VEBT_VEBT @ Ys @ N )
              = ( produc1678900780639429121T_VEBT @ P4 ) )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8397_in__set__zip,axiom,
    ! [P4: product_prod_int_nat,Xs: list_int,Ys: list_nat] :
      ( ( member216504246829706758nt_nat @ P4 @ ( set_Pr6647972299459129970nt_nat @ ( zip_int_nat @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_int @ Xs @ N )
              = ( product_fst_int_nat @ P4 ) )
            & ( ( nth_nat @ Ys @ N )
              = ( product_snd_int_nat @ P4 ) )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8398_in__set__zip,axiom,
    ! [P4: produc8025551001238799321T_VEBT,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( member8549952807677709168T_VEBT @ P4 @ ( set_Pr5984661752051438084T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_nat @ Xs @ N )
              = ( produc8575180428842422559T_VEBT @ P4 ) )
            & ( ( nth_VEBT_VEBT @ Ys @ N )
              = ( produc8172668247895388509T_VEBT @ P4 ) )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8399_in__set__zip,axiom,
    ! [P4: product_prod_nat_int,Xs: list_nat,Ys: list_int] :
      ( ( member4262671552274231302at_int @ P4 @ ( set_Pr1470767568048878706at_int @ ( zip_nat_int @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_nat @ Xs @ N )
              = ( product_fst_nat_int @ P4 ) )
            & ( ( nth_int @ Ys @ N )
              = ( product_snd_nat_int @ P4 ) )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8400_in__set__zip,axiom,
    ! [P4: product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ P4 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_nat @ Xs @ N )
              = ( product_fst_nat_nat @ P4 ) )
            & ( ( nth_nat @ Ys @ N )
              = ( product_snd_nat_nat @ P4 ) )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8401_in__set__zip,axiom,
    ! [P4: product_prod_int_int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ P4 @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
      = ( ? [N: nat] :
            ( ( ( nth_int @ Xs @ N )
              = ( product_fst_int_int @ P4 ) )
            & ( ( nth_int @ Ys @ N )
              = ( product_snd_int_int @ P4 ) )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_8402_size__prod__simp,axiom,
    ( basic_876126793109182934at_nat
    = ( ^ [F5: nat > nat,G2: nat > nat,P6: product_prod_nat_nat] : ( plus_plus_nat @ ( plus_plus_nat @ ( F5 @ ( product_fst_nat_nat @ P6 ) ) @ ( G2 @ ( product_snd_nat_nat @ P6 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_8403_size__prod__simp,axiom,
    ( basic_1872990034501187214nt_int
    = ( ^ [F5: int > nat,G2: int > nat,P6: product_prod_int_int] : ( plus_plus_nat @ ( plus_plus_nat @ ( F5 @ ( product_fst_int_int @ P6 ) ) @ ( G2 @ ( product_snd_int_int @ P6 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_8404_exI__realizer,axiom,
    ! [P2: product_prod_nat_nat > product_prod_nat_nat > $o,Y: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) ) @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_8405_exI__realizer,axiom,
    ! [P2: nat > vEBT_VEBT > $o,Y: nat,X: vEBT_VEBT] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc8311406018219409919BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) ) @ ( produc8713918199166443969BT_nat @ ( produc738532404422230701BT_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_8406_exI__realizer,axiom,
    ! [P2: extended_enat > vEBT_VEBT > $o,Y: extended_enat,X: vEBT_VEBT] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc522021263368680183d_enat @ ( produc581526299967858633d_enat @ X @ Y ) ) @ ( produc967593531271825845d_enat @ ( produc581526299967858633d_enat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_8407_exI__realizer,axiom,
    ! [P2: nat > nat > $o,Y: nat,X: nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_8408_exI__realizer,axiom,
    ! [P2: int > int > $o,Y: int,X: int] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_8409_conjI__realizer,axiom,
    ! [P2: product_prod_nat_nat > $o,P4: product_prod_nat_nat,Q: product_prod_nat_nat > $o,Q2: product_prod_nat_nat] :
      ( ( P2 @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc3213797794245857475at_nat @ ( produc6161850002892822231at_nat @ P4 @ Q2 ) ) )
          & ( Q @ ( produc6408287024330202629at_nat @ ( produc6161850002892822231at_nat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_8410_conjI__realizer,axiom,
    ! [P2: vEBT_VEBT > $o,P4: vEBT_VEBT,Q: nat > $o,Q2: nat] :
      ( ( P2 @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc8713918199166443969BT_nat @ ( produc738532404422230701BT_nat @ P4 @ Q2 ) ) )
          & ( Q @ ( produc8311406018219409919BT_nat @ ( produc738532404422230701BT_nat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_8411_conjI__realizer,axiom,
    ! [P2: vEBT_VEBT > $o,P4: vEBT_VEBT,Q: extended_enat > $o,Q2: extended_enat] :
      ( ( P2 @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc967593531271825845d_enat @ ( produc581526299967858633d_enat @ P4 @ Q2 ) ) )
          & ( Q @ ( produc522021263368680183d_enat @ ( produc581526299967858633d_enat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_8412_conjI__realizer,axiom,
    ! [P2: nat > $o,P4: nat,Q: nat > $o,Q2: nat] :
      ( ( P2 @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P4 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_8413_conjI__realizer,axiom,
    ! [P2: int > $o,P4: int,Q: int > $o,Q2: int] :
      ( ( P2 @ P4 )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_int_int @ ( product_Pair_int_int @ P4 @ Q2 ) ) )
          & ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P4 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_8414_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_8415_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X2: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_8416_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_8417_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_8418_distinct__union,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( distinct_int @ ( union_int @ Xs @ Ys ) )
      = ( distinct_int @ Ys ) ) ).

% distinct_union
thf(fact_8419_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_8420_set__remove1__eq,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ X @ Xs ) )
        = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT2 @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_remove1_eq
thf(fact_8421_set__remove1__eq,axiom,
    ! [Xs: list_real,X: real] :
      ( ( distinct_real @ Xs )
     => ( ( set_real2 @ ( remove1_real @ X @ Xs ) )
        = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real2 @ X @ bot_bot_set_real ) ) ) ) ).

% set_remove1_eq
thf(fact_8422_set__remove1__eq,axiom,
    ! [Xs: list_o,X: $o] :
      ( ( distinct_o @ Xs )
     => ( ( set_o2 @ ( remove1_o @ X @ Xs ) )
        = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o2 @ X @ bot_bot_set_o ) ) ) ) ).

% set_remove1_eq
thf(fact_8423_set__remove1__eq,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( ( set_int2 @ ( remove1_int @ X @ Xs ) )
        = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int2 @ X @ bot_bot_set_int ) ) ) ) ).

% set_remove1_eq
thf(fact_8424_set__remove1__eq,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( set_nat2 @ ( remove1_nat @ X @ Xs ) )
        = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_8425_root__powr__inverse,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_8426_horner__sum__eq__sum__funpow,axiom,
    ( groups6842663049115397189BT_int
    = ( ^ [F5: vEBT_VEBT > int,A3: int,Xs2: list_VEBT_VEBT] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( compow_int_int @ N @ ( times_times_int @ A3 ) @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8427_horner__sum__eq__sum__funpow,axiom,
    ( groups8485231416243008693nt_int
    = ( ^ [F5: int > int,A3: int,Xs2: list_int] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( compow_int_int @ N @ ( times_times_int @ A3 ) @ ( F5 @ ( nth_int @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8428_horner__sum__eq__sum__funpow,axiom,
    ( groups7485877704341954137at_int
    = ( ^ [F5: nat > int,A3: int,Xs2: list_nat] :
          ( groups3539618377306564664at_int
          @ ^ [N: nat] : ( compow_int_int @ N @ ( times_times_int @ A3 ) @ ( F5 @ ( nth_nat @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8429_horner__sum__eq__sum__funpow,axiom,
    ( groups1931381680841367751omplex
    = ( ^ [F5: vEBT_VEBT > complex,A3: complex,Xs2: list_VEBT_VEBT] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( compow6801810373992395016omplex @ N @ ( times_times_complex @ A3 ) @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8430_horner__sum__eq__sum__funpow,axiom,
    ( groups1380173120649922871omplex
    = ( ^ [F5: int > complex,A3: complex,Xs2: list_int] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( compow6801810373992395016omplex @ N @ ( times_times_complex @ A3 ) @ ( F5 @ ( nth_int @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8431_horner__sum__eq__sum__funpow,axiom,
    ( groups404637655443745499omplex
    = ( ^ [F5: nat > complex,A3: complex,Xs2: list_nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( compow6801810373992395016omplex @ N @ ( times_times_complex @ A3 ) @ ( F5 @ ( nth_nat @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8432_horner__sum__eq__sum__funpow,axiom,
    ( groups3844460828123498381d_enat
    = ( ^ [F5: vEBT_VEBT > extended_enat,A3: extended_enat,Xs2: list_VEBT_VEBT] :
          ( groups7108830773950497114d_enat
          @ ^ [N: nat] : ( compow4567540516116640754d_enat @ N @ ( times_7803423173614009249d_enat @ A3 ) @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8433_horner__sum__eq__sum__funpow,axiom,
    ( groups7888997386326813469d_enat
    = ( ^ [F5: int > extended_enat,A3: extended_enat,Xs2: list_int] :
          ( groups7108830773950497114d_enat
          @ ^ [N: nat] : ( compow4567540516116640754d_enat @ N @ ( times_7803423173614009249d_enat @ A3 ) @ ( F5 @ ( nth_int @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8434_horner__sum__eq__sum__funpow,axiom,
    ( groups1549203402269857401d_enat
    = ( ^ [F5: nat > extended_enat,A3: extended_enat,Xs2: list_nat] :
          ( groups7108830773950497114d_enat
          @ ^ [N: nat] : ( compow4567540516116640754d_enat @ N @ ( times_7803423173614009249d_enat @ A3 ) @ ( F5 @ ( nth_nat @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8435_horner__sum__eq__sum__funpow,axiom,
    ( groups6845153519624447465BT_nat
    = ( ^ [F5: vEBT_VEBT > nat,A3: nat,Xs2: list_VEBT_VEBT] :
          ( groups3542108847815614940at_nat
          @ ^ [N: nat] : ( compow_nat_nat @ N @ ( times_times_nat @ A3 ) @ ( F5 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% horner_sum_eq_sum_funpow
thf(fact_8436_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_8437_funpow__0,axiom,
    ! [F: nat > nat,X: nat] :
      ( ( compow_nat_nat @ zero_zero_nat @ F @ X )
      = X ) ).

% funpow_0
thf(fact_8438_in__set__remove1,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( ( A != B )
     => ( ( member_real2 @ A @ ( set_real2 @ ( remove1_real @ B @ Xs ) ) )
        = ( member_real2 @ A @ ( set_real2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_8439_in__set__remove1,axiom,
    ! [A: $o,B: $o,Xs: list_o] :
      ( ( A != B )
     => ( ( member_o2 @ A @ ( set_o2 @ ( remove1_o @ B @ Xs ) ) )
        = ( member_o2 @ A @ ( set_o2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_8440_in__set__remove1,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( ( A != B )
     => ( ( member_set_nat2 @ A @ ( set_set_nat2 @ ( remove1_set_nat @ B @ Xs ) ) )
        = ( member_set_nat2 @ A @ ( set_set_nat2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_8441_in__set__remove1,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( A != B )
     => ( ( member_VEBT_VEBT2 @ A @ ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ B @ Xs ) ) )
        = ( member_VEBT_VEBT2 @ A @ ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_8442_in__set__remove1,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( A != B )
     => ( ( member_int2 @ A @ ( set_int2 @ ( remove1_int @ B @ Xs ) ) )
        = ( member_int2 @ A @ ( set_int2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_8443_in__set__remove1,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( A != B )
     => ( ( member_nat2 @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs ) ) )
        = ( member_nat2 @ A @ ( set_nat2 @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_8444_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_8445_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_8446_real__root__eq__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = ( root @ N2 @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_8447_real__root__eq__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_8448_real__root__less__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_8449_real__root__le__iff,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_8450_real__root__eq__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_8451_real__root__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_8452_real__root__lt__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_8453_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_8454_real__root__le__0__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_8455_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_8456_real__root__lt__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_8457_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_8458_real__root__le__1__iff,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_8459_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_8460_real__root__pow__pos2,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_8461_funpow__swap1,axiom,
    ! [F: nat > nat,N2: nat,X: nat] :
      ( ( F @ ( compow_nat_nat @ N2 @ F @ X ) )
      = ( compow_nat_nat @ N2 @ F @ ( F @ X ) ) ) ).

% funpow_swap1
thf(fact_8462_funpow__mult,axiom,
    ! [N2: nat,M: nat,F: nat > nat] :
      ( ( compow_nat_nat @ N2 @ ( compow_nat_nat @ M @ F ) )
      = ( compow_nat_nat @ ( times_times_nat @ M @ N2 ) @ F ) ) ).

% funpow_mult
thf(fact_8463_notin__set__remove1,axiom,
    ! [X: real,Xs: list_real,Y: real] :
      ( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ~ ( member_real2 @ X @ ( set_real2 @ ( remove1_real @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_8464_notin__set__remove1,axiom,
    ! [X: $o,Xs: list_o,Y: $o] :
      ( ~ ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ~ ( member_o2 @ X @ ( set_o2 @ ( remove1_o @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_8465_notin__set__remove1,axiom,
    ! [X: set_nat,Xs: list_set_nat,Y: set_nat] :
      ( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ ( remove1_set_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_8466_notin__set__remove1,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ~ ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_8467_notin__set__remove1,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ~ ( member_int2 @ X @ ( set_int2 @ ( remove1_int @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_8468_notin__set__remove1,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ~ ( member_nat2 @ X @ ( set_nat2 @ ( remove1_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_8469_remove1__idem,axiom,
    ! [X: real,Xs: list_real] :
      ( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) )
     => ( ( remove1_real @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_8470_remove1__idem,axiom,
    ! [X: $o,Xs: list_o] :
      ( ~ ( member_o2 @ X @ ( set_o2 @ Xs ) )
     => ( ( remove1_o @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_8471_remove1__idem,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
     => ( ( remove1_set_nat @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_8472_remove1__idem,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( remove1_VEBT_VEBT @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_8473_remove1__idem,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( remove1_int @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_8474_remove1__idem,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( remove1_nat @ X @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_8475_distinct__remove1,axiom,
    ! [Xs: list_int,X: int] :
      ( ( distinct_int @ Xs )
     => ( distinct_int @ ( remove1_int @ X @ Xs ) ) ) ).

% distinct_remove1
thf(fact_8476_distinct__remove1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( remove1_nat @ X @ Xs ) ) ) ).

% distinct_remove1
thf(fact_8477_set__remove1__subset,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( remove1_VEBT_VEBT @ X @ Xs ) ) @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_8478_set__remove1__subset,axiom,
    ! [X: int,Xs: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( remove1_int @ X @ Xs ) ) @ ( set_int2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_8479_set__remove1__subset,axiom,
    ! [X: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_remove1_subset
thf(fact_8480_real__root__less__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_8481_real__root__le__mono,axiom,
    ! [N2: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_8482_real__root__power,axiom,
    ! [N2: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_8483_real__root__abs,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_abs
thf(fact_8484_sgn__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_8485_of__nat__def,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] : ( compow6801810373992395016omplex @ N @ ( plus_plus_complex @ one_one_complex ) @ zero_zero_complex ) ) ) ).

% of_nat_def
thf(fact_8486_of__nat__def,axiom,
    ( semiri4216267220026989637d_enat
    = ( ^ [N: nat] : ( compow4567540516116640754d_enat @ N @ ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat ) @ zero_z5237406670263579293d_enat ) ) ) ).

% of_nat_def
thf(fact_8487_of__nat__def,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] : ( compow_real_real @ N @ ( plus_plus_real @ one_one_real ) @ zero_zero_real ) ) ) ).

% of_nat_def
thf(fact_8488_of__nat__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] : ( compow_int_int @ N @ ( plus_plus_int @ one_one_int ) @ zero_zero_int ) ) ) ).

% of_nat_def
thf(fact_8489_of__nat__def,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : ( compow_nat_nat @ N @ ( plus_plus_nat @ one_one_nat ) @ zero_zero_nat ) ) ) ).

% of_nat_def
thf(fact_8490_real__root__gt__zero,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_8491_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ K ) @ A )
      = ( compow6801810373992395016omplex @ ( numeral_numeral_nat @ K ) @ ( plus_plus_complex @ one_one_complex ) @ A ) ) ).

% numeral_add_unfold_funpow
thf(fact_8492_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ K ) @ A )
      = ( compow_nat_nat @ ( numeral_numeral_nat @ K ) @ ( plus_plus_nat @ one_one_nat ) @ A ) ) ).

% numeral_add_unfold_funpow
thf(fact_8493_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ K ) @ A )
      = ( compow4567540516116640754d_enat @ ( numeral_numeral_nat @ K ) @ ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat ) @ A ) ) ).

% numeral_add_unfold_funpow
thf(fact_8494_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ K ) @ A )
      = ( compow_int_int @ ( numeral_numeral_nat @ K ) @ ( plus_plus_int @ one_one_int ) @ A ) ) ).

% numeral_add_unfold_funpow
thf(fact_8495_numeral__add__unfold__funpow,axiom,
    ! [K: num,A: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ K ) @ A )
      = ( compow_real_real @ ( numeral_numeral_nat @ K ) @ ( plus_plus_real @ one_one_real ) @ A ) ) ).

% numeral_add_unfold_funpow
thf(fact_8496_real__root__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N7 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N7 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_8497_root__abs__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_8498_numeral__unfold__funpow,axiom,
    ( numera6690914467698888265omplex
    = ( ^ [K2: num] : ( compow6801810373992395016omplex @ ( numeral_numeral_nat @ K2 ) @ ( plus_plus_complex @ one_one_complex ) @ zero_zero_complex ) ) ) ).

% numeral_unfold_funpow
thf(fact_8499_numeral__unfold__funpow,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( compow_nat_nat @ ( numeral_numeral_nat @ K2 ) @ ( plus_plus_nat @ one_one_nat ) @ zero_zero_nat ) ) ) ).

% numeral_unfold_funpow
thf(fact_8500_numeral__unfold__funpow,axiom,
    ( numera1916890842035813515d_enat
    = ( ^ [K2: num] : ( compow4567540516116640754d_enat @ ( numeral_numeral_nat @ K2 ) @ ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat ) @ zero_z5237406670263579293d_enat ) ) ) ).

% numeral_unfold_funpow
thf(fact_8501_numeral__unfold__funpow,axiom,
    ( numeral_numeral_int
    = ( ^ [K2: num] : ( compow_int_int @ ( numeral_numeral_nat @ K2 ) @ ( plus_plus_int @ one_one_int ) @ zero_zero_int ) ) ) ).

% numeral_unfold_funpow
thf(fact_8502_numeral__unfold__funpow,axiom,
    ( numeral_numeral_real
    = ( ^ [K2: num] : ( compow_real_real @ ( numeral_numeral_nat @ K2 ) @ ( plus_plus_real @ one_one_real ) @ zero_zero_real ) ) ) ).

% numeral_unfold_funpow
thf(fact_8503_real__root__pos__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_8504_real__root__strict__increasing,axiom,
    ! [N2: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N7 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_8505_real__root__decreasing,axiom,
    ! [N2: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N7 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N7 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_8506_real__root__pow__pos,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_8507_real__root__pos__unique,axiom,
    ! [N2: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N2 )
            = X )
         => ( ( root @ N2 @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_8508_real__root__power__cancel,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_8509_length__remove1,axiom,
    ! [X: real,Xs: list_real] :
      ( ( ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ( ( size_size_list_real @ ( remove1_real @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_real @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_real2 @ X @ ( set_real2 @ Xs ) )
       => ( ( size_size_list_real @ ( remove1_real @ X @ Xs ) )
          = ( size_size_list_real @ Xs ) ) ) ) ).

% length_remove1
thf(fact_8510_length__remove1,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ( ( size_size_list_o @ ( remove1_o @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_o @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_o2 @ X @ ( set_o2 @ Xs ) )
       => ( ( size_size_list_o @ ( remove1_o @ X @ Xs ) )
          = ( size_size_list_o @ Xs ) ) ) ) ).

% length_remove1
thf(fact_8511_length__remove1,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
       => ( ( size_s3254054031482475050et_nat @ ( remove1_set_nat @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat2 @ X @ ( set_set_nat2 @ Xs ) )
       => ( ( size_s3254054031482475050et_nat @ ( remove1_set_nat @ X @ Xs ) )
          = ( size_s3254054031482475050et_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_8512_length__remove1,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( ( size_s6755466524823107622T_VEBT @ ( remove1_VEBT_VEBT @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_VEBT_VEBT2 @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( ( size_s6755466524823107622T_VEBT @ ( remove1_VEBT_VEBT @ X @ Xs ) )
          = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).

% length_remove1
thf(fact_8513_length__remove1,axiom,
    ! [X: int,Xs: list_int] :
      ( ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ( ( size_size_list_int @ ( remove1_int @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ( ( size_size_list_int @ ( remove1_int @ X @ Xs ) )
          = ( size_size_list_int @ Xs ) ) ) ) ).

% length_remove1
thf(fact_8514_length__remove1,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
          = ( size_size_list_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_8515_real__root__increasing,axiom,
    ! [N2: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N7 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_8516_sgn__power__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
        = X ) ) ).

% sgn_power_root
thf(fact_8517_root__sgn__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_8518_ln__root,axiom,
    ! [N2: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N2 @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_8519_log__root,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N2 @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_8520_log__base__root,axiom,
    ! [N2: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_8521_split__root,axiom,
    ! [P2: real > $o,N2: nat,X: real] :
      ( ( P2 @ ( root @ N2 @ X ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P2 @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
                = X )
             => ( P2 @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_8522_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_8523_Nat_Ofunpow__code__def,axiom,
    funpow_nat = compow_nat_nat ).

% Nat.funpow_code_def
thf(fact_8524_polyfun__rootbound,axiom,
    ! [C: nat > complex,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( finite3207457112153483333omplex
            @ ( collect_complex
              @ ^ [Z6: complex] :
                  ( ( groups2073611262835488442omplex
                    @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ Z6 @ I5 ) )
                    @ ( set_ord_atMost_nat @ N2 ) )
                  = zero_zero_complex ) ) )
          & ( ord_less_eq_nat
            @ ( finite_card_complex
              @ ( collect_complex
                @ ^ [Z6: complex] :
                    ( ( groups2073611262835488442omplex
                      @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ Z6 @ I5 ) )
                      @ ( set_ord_atMost_nat @ N2 ) )
                    = zero_zero_complex ) ) )
            @ N2 ) ) ) ) ).

% polyfun_rootbound
thf(fact_8525_polyfun__rootbound,axiom,
    ! [C: nat > real,K: nat,N2: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( finite_finite_real
            @ ( collect_real
              @ ^ [Z6: real] :
                  ( ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ Z6 @ I5 ) )
                    @ ( set_ord_atMost_nat @ N2 ) )
                  = zero_zero_real ) ) )
          & ( ord_less_eq_nat
            @ ( finite_card_real
              @ ( collect_real
                @ ^ [Z6: real] :
                    ( ( groups6591440286371151544t_real
                      @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ Z6 @ I5 ) )
                      @ ( set_ord_atMost_nat @ N2 ) )
                    = zero_zero_real ) ) )
            @ N2 ) ) ) ) ).

% polyfun_rootbound
thf(fact_8526_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] : ( ord_less_nat @ I5 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_8527_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_8528_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] : ( ord_less_eq_nat @ I5 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_8529_card_Oempty,axiom,
    ( ( finite_card_complex @ bot_bot_set_complex )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8530_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8531_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8532_card_Oempty,axiom,
    ( ( finite_card_real @ bot_bot_set_real )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8533_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8534_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8535_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_8536_card_Oinfinite,axiom,
    ! [A2: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite_card_set_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_8537_card_Oinfinite,axiom,
    ! [A2: set_list_nat] :
      ( ~ ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite_card_list_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_8538_card_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_card_nat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_8539_card_Oinfinite,axiom,
    ! [A2: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_card_complex @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_8540_card_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_card_int @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_8541_card_Oinfinite,axiom,
    ! [A2: set_Extended_enat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite121521170596916366d_enat @ A2 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_8542_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_8543_card__0__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( ( finite_card_set_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_8544_card__0__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( ( finite_card_list_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
thf(fact_8545_card__0__eq,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( finite_card_complex @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_complex ) ) ) ).

% card_0_eq
thf(fact_8546_card__0__eq,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( finite121521170596916366d_enat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bo7653980558646680370d_enat ) ) ) ).

% card_0_eq
thf(fact_8547_card__0__eq,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( finite_card_real @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_real ) ) ) ).

% card_0_eq
thf(fact_8548_card__0__eq,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( finite_card_o @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_8549_card__0__eq,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( finite_card_nat @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_8550_card__0__eq,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( finite_card_int @ A2 )
          = zero_zero_nat )
        = ( A2 = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_8551_card__insert__disjoint,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real2 @ X @ A2 )
       => ( ( finite_card_real @ ( insert_real2 @ X @ A2 ) )
          = ( suc @ ( finite_card_real @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8552_card__insert__disjoint,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o2 @ X @ A2 )
       => ( ( finite_card_o @ ( insert_o2 @ X @ A2 ) )
          = ( suc @ ( finite_card_o @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8553_card__insert__disjoint,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ~ ( member_set_nat2 @ X @ A2 )
       => ( ( finite_card_set_nat @ ( insert_set_nat2 @ X @ A2 ) )
          = ( suc @ ( finite_card_set_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8554_card__insert__disjoint,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ~ ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
          = ( suc @ ( finite_card_list_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8555_card__insert__disjoint,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat2 @ X @ A2 )
       => ( ( finite_card_nat @ ( insert_nat2 @ X @ A2 ) )
          = ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8556_card__insert__disjoint,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
          = ( suc @ ( finite_card_complex @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8557_card__insert__disjoint,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int2 @ X @ A2 )
       => ( ( finite_card_int @ ( insert_int2 @ X @ A2 ) )
          = ( suc @ ( finite_card_int @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8558_card__insert__disjoint,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ~ ( member_Extended_enat @ X @ A2 )
       => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
          = ( suc @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_8559_n__subsets,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite1149291290879098388et_nat
          @ ( collect_set_set_nat
            @ ^ [B5: set_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ B5 @ A2 )
                & ( ( finite_card_set_nat @ B5 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_set_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8560_n__subsets,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite2364142230527598318st_nat
          @ ( collect_set_list_nat
            @ ^ [B5: set_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ B5 @ A2 )
                & ( ( finite_card_list_nat @ B5 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_list_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8561_n__subsets,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite903997441450111292omplex
          @ ( collect_set_complex
            @ ^ [B5: set_complex] :
                ( ( ord_le211207098394363844omplex @ B5 @ A2 )
                & ( ( finite_card_complex @ B5 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_complex @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8562_n__subsets,axiom,
    ! [A2: set_int,K: nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_card_set_int
          @ ( collect_set_int
            @ ^ [B5: set_int] :
                ( ( ord_less_eq_set_int @ B5 @ A2 )
                & ( ( finite_card_int @ B5 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_int @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8563_n__subsets,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite3719263829065406702d_enat
          @ ( collec2260605976452661553d_enat
            @ ^ [B5: set_Extended_enat] :
                ( ( ord_le7203529160286727270d_enat @ B5 @ A2 )
                & ( ( finite121521170596916366d_enat @ B5 )
                  = K ) ) ) )
        = ( binomial @ ( finite121521170596916366d_enat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8564_n__subsets,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [B5: set_nat] :
                ( ( ord_less_eq_set_nat @ B5 @ A2 )
                & ( ( finite_card_nat @ B5 )
                  = K ) ) ) )
        = ( binomial @ ( finite_card_nat @ A2 ) @ K ) ) ) ).

% n_subsets
thf(fact_8565_infinite__arbitrarily__large,axiom,
    ! [A2: set_set_nat,N2: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A2 )
     => ? [B7: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B7 )
          & ( ( finite_card_set_nat @ B7 )
            = N2 )
          & ( ord_le6893508408891458716et_nat @ B7 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_8566_infinite__arbitrarily__large,axiom,
    ! [A2: set_list_nat,N2: nat] :
      ( ~ ( finite8100373058378681591st_nat @ A2 )
     => ? [B7: set_list_nat] :
          ( ( finite8100373058378681591st_nat @ B7 )
          & ( ( finite_card_list_nat @ B7 )
            = N2 )
          & ( ord_le6045566169113846134st_nat @ B7 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_8567_infinite__arbitrarily__large,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ? [B7: set_complex] :
          ( ( finite3207457112153483333omplex @ B7 )
          & ( ( finite_card_complex @ B7 )
            = N2 )
          & ( ord_le211207098394363844omplex @ B7 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_8568_infinite__arbitrarily__large,axiom,
    ! [A2: set_int,N2: nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ? [B7: set_int] :
          ( ( finite_finite_int @ B7 )
          & ( ( finite_card_int @ B7 )
            = N2 )
          & ( ord_less_eq_set_int @ B7 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_8569_infinite__arbitrarily__large,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ~ ( finite4001608067531595151d_enat @ A2 )
     => ? [B7: set_Extended_enat] :
          ( ( finite4001608067531595151d_enat @ B7 )
          & ( ( finite121521170596916366d_enat @ B7 )
            = N2 )
          & ( ord_le7203529160286727270d_enat @ B7 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_8570_infinite__arbitrarily__large,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ? [B7: set_nat] :
          ( ( finite_finite_nat @ B7 )
          & ( ( finite_card_nat @ B7 )
            = N2 )
          & ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_8571_card__subset__eq,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ( ( finite_card_set_nat @ A2 )
            = ( finite_card_set_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_8572_card__subset__eq,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ( ( finite_card_list_nat @ A2 )
            = ( finite_card_list_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_8573_card__subset__eq,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( ( finite_card_complex @ A2 )
            = ( finite_card_complex @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_8574_card__subset__eq,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( ( finite_card_int @ A2 )
            = ( finite_card_int @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_8575_card__subset__eq,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( ( finite121521170596916366d_enat @ A2 )
            = ( finite121521170596916366d_enat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_8576_card__subset__eq,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( ( finite_card_nat @ A2 )
            = ( finite_card_nat @ B2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_subset_eq
thf(fact_8577_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_real,R2: real > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A5: real] :
            ( ( member_real2 @ A5 @ A2 )
           => ? [B8: real] :
                ( ( member_real2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: real,A24: real,B4: real] :
              ( ( member_real2 @ A13 @ A2 )
             => ( ( member_real2 @ A24 @ A2 )
               => ( ( member_real2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8578_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_real,R2: real > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: real] :
            ( ( member_real2 @ A5 @ A2 )
           => ? [B8: $o] :
                ( ( member_o2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: real,A24: real,B4: $o] :
              ( ( member_real2 @ A13 @ A2 )
             => ( ( member_real2 @ A24 @ A2 )
               => ( ( member_o2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8579_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_o,R2: $o > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A5: $o] :
            ( ( member_o2 @ A5 @ A2 )
           => ? [B8: real] :
                ( ( member_real2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: $o,A24: $o,B4: real] :
              ( ( member_o2 @ A13 @ A2 )
             => ( ( member_o2 @ A24 @ A2 )
               => ( ( member_real2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8580_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_o,R2: $o > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: $o] :
            ( ( member_o2 @ A5 @ A2 )
           => ? [B8: $o] :
                ( ( member_o2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: $o,A24: $o,B4: $o] :
              ( ( member_o2 @ A13 @ A2 )
             => ( ( member_o2 @ A24 @ A2 )
               => ( ( member_o2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8581_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_int,R2: int > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A5: int] :
            ( ( member_int2 @ A5 @ A2 )
           => ? [B8: real] :
                ( ( member_real2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: int,A24: int,B4: real] :
              ( ( member_int2 @ A13 @ A2 )
             => ( ( member_int2 @ A24 @ A2 )
               => ( ( member_real2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8582_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_int,R2: int > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: int] :
            ( ( member_int2 @ A5 @ A2 )
           => ? [B8: $o] :
                ( ( member_o2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: int,A24: int,B4: $o] :
              ( ( member_int2 @ A13 @ A2 )
             => ( ( member_int2 @ A24 @ A2 )
               => ( ( member_o2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8583_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_nat,R2: nat > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A5: nat] :
            ( ( member_nat2 @ A5 @ A2 )
           => ? [B8: real] :
                ( ( member_real2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: nat,A24: nat,B4: real] :
              ( ( member_nat2 @ A13 @ A2 )
             => ( ( member_nat2 @ A24 @ A2 )
               => ( ( member_real2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8584_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_nat,R2: nat > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: nat] :
            ( ( member_nat2 @ A5 @ A2 )
           => ? [B8: $o] :
                ( ( member_o2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: nat,A24: nat,B4: $o] :
              ( ( member_nat2 @ A13 @ A2 )
             => ( ( member_nat2 @ A24 @ A2 )
               => ( ( member_o2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8585_card__le__if__inj__on__rel,axiom,
    ! [B2: set_real,A2: set_complex,R2: complex > real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ? [B8: real] :
                ( ( member_real2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: complex,A24: complex,B4: real] :
              ( ( member_complex @ A13 @ A2 )
             => ( ( member_complex @ A24 @ A2 )
               => ( ( member_real2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8586_card__le__if__inj__on__rel,axiom,
    ! [B2: set_o,A2: set_complex,R2: complex > $o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ? [B8: $o] :
                ( ( member_o2 @ B8 @ B2 )
                & ( R2 @ A5 @ B8 ) ) )
       => ( ! [A13: complex,A24: complex,B4: $o] :
              ( ( member_complex @ A13 @ A2 )
             => ( ( member_complex @ A24 @ A2 )
               => ( ( member_o2 @ B4 @ B2 )
                 => ( ( R2 @ A13 @ B4 )
                   => ( ( R2 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_8587_card__insert__le,axiom,
    ! [A2: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ ( insert_int2 @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8588_card__insert__le,axiom,
    ! [A2: set_real,X: real] : ( ord_less_eq_nat @ ( finite_card_real @ A2 ) @ ( finite_card_real @ ( insert_real2 @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8589_card__insert__le,axiom,
    ! [A2: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_o @ ( insert_o2 @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8590_card__insert__le,axiom,
    ! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat2 @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8591_card__insert__le,axiom,
    ! [A2: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ ( insert_complex @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8592_card__insert__le,axiom,
    ! [A2: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ ( insert_set_nat2 @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8593_card__insert__le,axiom,
    ! [A2: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) ) ) ).

% card_insert_le
thf(fact_8594_card__lists__length__eq,axiom,
    ! [A2: set_set_nat,N2: nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs2: list_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A2 )
                & ( ( size_s3254054031482475050et_nat @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite_card_set_nat @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8595_card__lists__length__eq,axiom,
    ! [A2: set_list_nat,N2: nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs2: list_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A2 )
                & ( ( size_s3023201423986296836st_nat @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite_card_list_nat @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8596_card__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs2: list_complex] :
                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
                & ( ( size_s3451745648224563538omplex @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite_card_complex @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8597_card__lists__length__eq,axiom,
    ! [A2: set_Extended_enat,N2: nat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite7441382602597825044d_enat
          @ ( collec8433460942617342167d_enat
            @ ^ [Xs2: list_Extended_enat] :
                ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A2 )
                & ( ( size_s3941691890525107288d_enat @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite121521170596916366d_enat @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8598_card__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs2: list_VEBT_VEBT] :
                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8599_card__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_card_list_int
          @ ( collect_list_int
            @ ^ [Xs2: list_int] :
                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
                & ( ( size_size_list_int @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite_card_int @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8600_card__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [Xs2: list_nat] :
                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
                & ( ( size_size_list_nat @ Xs2 )
                  = N2 ) ) ) )
        = ( power_power_nat @ ( finite_card_nat @ A2 ) @ N2 ) ) ) ).

% card_lists_length_eq
thf(fact_8601_card__eq__0__iff,axiom,
    ! [A2: set_set_nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_set_nat )
        | ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8602_card__eq__0__iff,axiom,
    ! [A2: set_list_nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_list_nat )
        | ~ ( finite8100373058378681591st_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8603_card__eq__0__iff,axiom,
    ! [A2: set_complex] :
      ( ( ( finite_card_complex @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_complex )
        | ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8604_card__eq__0__iff,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ( finite121521170596916366d_enat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bo7653980558646680370d_enat )
        | ~ ( finite4001608067531595151d_enat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8605_card__eq__0__iff,axiom,
    ! [A2: set_real] :
      ( ( ( finite_card_real @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_real )
        | ~ ( finite_finite_real @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8606_card__eq__0__iff,axiom,
    ! [A2: set_o] :
      ( ( ( finite_card_o @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_o )
        | ~ ( finite_finite_o @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8607_card__eq__0__iff,axiom,
    ! [A2: set_nat] :
      ( ( ( finite_card_nat @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8608_card__eq__0__iff,axiom,
    ! [A2: set_int] :
      ( ( ( finite_card_int @ A2 )
        = zero_zero_nat )
      = ( ( A2 = bot_bot_set_int )
        | ~ ( finite_finite_int @ A2 ) ) ) ).

% card_eq_0_iff
thf(fact_8609_card__ge__0__finite,axiom,
    ! [A2: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
     => ( finite1152437895449049373et_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_8610_card__ge__0__finite,axiom,
    ! [A2: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A2 ) )
     => ( finite8100373058378681591st_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_8611_card__ge__0__finite,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
     => ( finite_finite_nat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_8612_card__ge__0__finite,axiom,
    ! [A2: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
     => ( finite3207457112153483333omplex @ A2 ) ) ).

% card_ge_0_finite
thf(fact_8613_card__ge__0__finite,axiom,
    ! [A2: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
     => ( finite_finite_int @ A2 ) ) ).

% card_ge_0_finite
thf(fact_8614_card__ge__0__finite,axiom,
    ! [A2: set_Extended_enat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A2 ) )
     => ( finite4001608067531595151d_enat @ A2 ) ) ).

% card_ge_0_finite
thf(fact_8615_card__Suc__eq__finite,axiom,
    ! [A2: set_real,K: nat] :
      ( ( ( finite_card_real @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: real,B5: set_real] :
            ( ( A2
              = ( insert_real2 @ B3 @ B5 ) )
            & ~ ( member_real2 @ B3 @ B5 )
            & ( ( finite_card_real @ B5 )
              = K )
            & ( finite_finite_real @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8616_card__Suc__eq__finite,axiom,
    ! [A2: set_o,K: nat] :
      ( ( ( finite_card_o @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: $o,B5: set_o] :
            ( ( A2
              = ( insert_o2 @ B3 @ B5 ) )
            & ~ ( member_o2 @ B3 @ B5 )
            & ( ( finite_card_o @ B5 )
              = K )
            & ( finite_finite_o @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8617_card__Suc__eq__finite,axiom,
    ! [A2: set_set_nat,K: nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: set_nat,B5: set_set_nat] :
            ( ( A2
              = ( insert_set_nat2 @ B3 @ B5 ) )
            & ~ ( member_set_nat2 @ B3 @ B5 )
            & ( ( finite_card_set_nat @ B5 )
              = K )
            & ( finite1152437895449049373et_nat @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8618_card__Suc__eq__finite,axiom,
    ! [A2: set_list_nat,K: nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: list_nat,B5: set_list_nat] :
            ( ( A2
              = ( insert_list_nat @ B3 @ B5 ) )
            & ~ ( member_list_nat @ B3 @ B5 )
            & ( ( finite_card_list_nat @ B5 )
              = K )
            & ( finite8100373058378681591st_nat @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8619_card__Suc__eq__finite,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ( finite_card_nat @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: nat,B5: set_nat] :
            ( ( A2
              = ( insert_nat2 @ B3 @ B5 ) )
            & ~ ( member_nat2 @ B3 @ B5 )
            & ( ( finite_card_nat @ B5 )
              = K )
            & ( finite_finite_nat @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8620_card__Suc__eq__finite,axiom,
    ! [A2: set_complex,K: nat] :
      ( ( ( finite_card_complex @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: complex,B5: set_complex] :
            ( ( A2
              = ( insert_complex @ B3 @ B5 ) )
            & ~ ( member_complex @ B3 @ B5 )
            & ( ( finite_card_complex @ B5 )
              = K )
            & ( finite3207457112153483333omplex @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8621_card__Suc__eq__finite,axiom,
    ! [A2: set_int,K: nat] :
      ( ( ( finite_card_int @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: int,B5: set_int] :
            ( ( A2
              = ( insert_int2 @ B3 @ B5 ) )
            & ~ ( member_int2 @ B3 @ B5 )
            & ( ( finite_card_int @ B5 )
              = K )
            & ( finite_finite_int @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8622_card__Suc__eq__finite,axiom,
    ! [A2: set_Extended_enat,K: nat] :
      ( ( ( finite121521170596916366d_enat @ A2 )
        = ( suc @ K ) )
      = ( ? [B3: extended_enat,B5: set_Extended_enat] :
            ( ( A2
              = ( insert_Extended_enat @ B3 @ B5 ) )
            & ~ ( member_Extended_enat @ B3 @ B5 )
            & ( ( finite121521170596916366d_enat @ B5 )
              = K )
            & ( finite4001608067531595151d_enat @ B5 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_8623_card__insert__if,axiom,
    ! [A2: set_real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real2 @ X @ A2 )
         => ( ( finite_card_real @ ( insert_real2 @ X @ A2 ) )
            = ( finite_card_real @ A2 ) ) )
        & ( ~ ( member_real2 @ X @ A2 )
         => ( ( finite_card_real @ ( insert_real2 @ X @ A2 ) )
            = ( suc @ ( finite_card_real @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8624_card__insert__if,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o2 @ X @ A2 )
         => ( ( finite_card_o @ ( insert_o2 @ X @ A2 ) )
            = ( finite_card_o @ A2 ) ) )
        & ( ~ ( member_o2 @ X @ A2 )
         => ( ( finite_card_o @ ( insert_o2 @ X @ A2 ) )
            = ( suc @ ( finite_card_o @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8625_card__insert__if,axiom,
    ! [A2: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( ( member_set_nat2 @ X @ A2 )
         => ( ( finite_card_set_nat @ ( insert_set_nat2 @ X @ A2 ) )
            = ( finite_card_set_nat @ A2 ) ) )
        & ( ~ ( member_set_nat2 @ X @ A2 )
         => ( ( finite_card_set_nat @ ( insert_set_nat2 @ X @ A2 ) )
            = ( suc @ ( finite_card_set_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8626_card__insert__if,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( ( member_list_nat @ X @ A2 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
            = ( finite_card_list_nat @ A2 ) ) )
        & ( ~ ( member_list_nat @ X @ A2 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) )
            = ( suc @ ( finite_card_list_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8627_card__insert__if,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat2 @ X @ A2 )
         => ( ( finite_card_nat @ ( insert_nat2 @ X @ A2 ) )
            = ( finite_card_nat @ A2 ) ) )
        & ( ~ ( member_nat2 @ X @ A2 )
         => ( ( finite_card_nat @ ( insert_nat2 @ X @ A2 ) )
            = ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8628_card__insert__if,axiom,
    ! [A2: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
            = ( finite_card_complex @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A2 ) )
            = ( suc @ ( finite_card_complex @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8629_card__insert__if,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int2 @ X @ A2 )
         => ( ( finite_card_int @ ( insert_int2 @ X @ A2 ) )
            = ( finite_card_int @ A2 ) ) )
        & ( ~ ( member_int2 @ X @ A2 )
         => ( ( finite_card_int @ ( insert_int2 @ X @ A2 ) )
            = ( suc @ ( finite_card_int @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8630_card__insert__if,axiom,
    ! [A2: set_Extended_enat,X: extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( ( member_Extended_enat @ X @ A2 )
         => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
            = ( finite121521170596916366d_enat @ A2 ) ) )
        & ( ~ ( member_Extended_enat @ X @ A2 )
         => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X @ A2 ) )
            = ( suc @ ( finite121521170596916366d_enat @ A2 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_8631_card__mono,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_8632_card__mono,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_8633_card__mono,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) ) ) ) ).

% card_mono
thf(fact_8634_card__mono,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ).

% card_mono
thf(fact_8635_card__mono,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).

% card_mono
thf(fact_8636_card__mono,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).

% card_mono
thf(fact_8637_card__seteq,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_8638_card__seteq,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B2 ) @ ( finite_card_list_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_8639_card__seteq,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ B2 ) @ ( finite_card_complex @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_8640_card__seteq,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_8641_card__seteq,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le7203529160286727270d_enat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ B2 ) @ ( finite121521170596916366d_enat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_8642_card__seteq,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
thf(fact_8643_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S3: set_set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ S3 ) )
     => ~ ! [T5: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T5 @ S3 )
           => ( ( ( finite_card_set_nat @ T5 )
                = N2 )
             => ~ ( finite1152437895449049373et_nat @ T5 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_8644_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S3: set_list_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ S3 ) )
     => ~ ! [T5: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T5 @ S3 )
           => ( ( ( finite_card_list_nat @ T5 )
                = N2 )
             => ~ ( finite8100373058378681591st_nat @ T5 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_8645_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S3: set_complex] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_complex @ S3 ) )
     => ~ ! [T5: set_complex] :
            ( ( ord_le211207098394363844omplex @ T5 @ S3 )
           => ( ( ( finite_card_complex @ T5 )
                = N2 )
             => ~ ( finite3207457112153483333omplex @ T5 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_8646_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S3: set_int] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S3 ) )
     => ~ ! [T5: set_int] :
            ( ( ord_less_eq_set_int @ T5 @ S3 )
           => ( ( ( finite_card_int @ T5 )
                = N2 )
             => ~ ( finite_finite_int @ T5 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_8647_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S3: set_Extended_enat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite121521170596916366d_enat @ S3 ) )
     => ~ ! [T5: set_Extended_enat] :
            ( ( ord_le7203529160286727270d_enat @ T5 @ S3 )
           => ( ( ( finite121521170596916366d_enat @ T5 )
                = N2 )
             => ~ ( finite4001608067531595151d_enat @ T5 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_8648_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S3: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S3 ) )
     => ~ ! [T5: set_nat] :
            ( ( ord_less_eq_set_nat @ T5 @ S3 )
           => ( ( ( finite_card_nat @ T5 )
                = N2 )
             => ~ ( finite_finite_nat @ T5 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_8649_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_set_nat,C4: nat] :
      ( ! [G3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G3 @ F3 )
         => ( ( finite1152437895449049373et_nat @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G3 ) @ C4 ) ) )
     => ( ( finite1152437895449049373et_nat @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_8650_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_list_nat,C4: nat] :
      ( ! [G3: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G3 @ F3 )
         => ( ( finite8100373058378681591st_nat @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G3 ) @ C4 ) ) )
     => ( ( finite8100373058378681591st_nat @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_8651_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_complex,C4: nat] :
      ( ! [G3: set_complex] :
          ( ( ord_le211207098394363844omplex @ G3 @ F3 )
         => ( ( finite3207457112153483333omplex @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_complex @ G3 ) @ C4 ) ) )
     => ( ( finite3207457112153483333omplex @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_complex @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_8652_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_int,C4: nat] :
      ( ! [G3: set_int] :
          ( ( ord_less_eq_set_int @ G3 @ F3 )
         => ( ( finite_finite_int @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_int @ G3 ) @ C4 ) ) )
     => ( ( finite_finite_int @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_8653_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_Extended_enat,C4: nat] :
      ( ! [G3: set_Extended_enat] :
          ( ( ord_le7203529160286727270d_enat @ G3 @ F3 )
         => ( ( finite4001608067531595151d_enat @ G3 )
           => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ G3 ) @ C4 ) ) )
     => ( ( finite4001608067531595151d_enat @ F3 )
        & ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_8654_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_nat,C4: nat] :
      ( ! [G3: set_nat] :
          ( ( ord_less_eq_set_nat @ G3 @ F3 )
         => ( ( finite_finite_nat @ G3 )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C4 ) ) )
     => ( ( finite_finite_nat @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_8655_card__less__sym__Diff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_8656_card__less__sym__Diff,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_8657_card__less__sym__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ord_less_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_8658_card__less__sym__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_8659_card__less__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_8660_card__less__sym__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_less_sym_Diff
thf(fact_8661_card__le__sym__Diff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_8662_card__le__sym__Diff,axiom,
    ! [A2: set_list_nat,B2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ( finite8100373058378681591st_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_8663_card__le__sym__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_8664_card__le__sym__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_8665_card__le__sym__Diff,axiom,
    ! [A2: set_Extended_enat,B2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( finite4001608067531595151d_enat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A2 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_8666_card__le__sym__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_8667_card__length,axiom,
    ! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).

% card_length
thf(fact_8668_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_8669_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_8670_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_8671_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_8672_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_8673_card__1__singletonE,axiom,
    ! [A2: set_complex] :
      ( ( ( finite_card_complex @ A2 )
        = one_one_nat )
     => ~ ! [X5: complex] :
            ( A2
           != ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ).

% card_1_singletonE
thf(fact_8674_card__1__singletonE,axiom,
    ! [A2: set_set_nat] :
      ( ( ( finite_card_set_nat @ A2 )
        = one_one_nat )
     => ~ ! [X5: set_nat] :
            ( A2
           != ( insert_set_nat2 @ X5 @ bot_bot_set_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_8675_card__1__singletonE,axiom,
    ! [A2: set_list_nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = one_one_nat )
     => ~ ! [X5: list_nat] :
            ( A2
           != ( insert_list_nat @ X5 @ bot_bot_set_list_nat ) ) ) ).

% card_1_singletonE
thf(fact_8676_card__1__singletonE,axiom,
    ! [A2: set_real] :
      ( ( ( finite_card_real @ A2 )
        = one_one_nat )
     => ~ ! [X5: real] :
            ( A2
           != ( insert_real2 @ X5 @ bot_bot_set_real ) ) ) ).

% card_1_singletonE
thf(fact_8677_card__1__singletonE,axiom,
    ! [A2: set_o] :
      ( ( ( finite_card_o @ A2 )
        = one_one_nat )
     => ~ ! [X5: $o] :
            ( A2
           != ( insert_o2 @ X5 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_8678_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( ( finite_card_nat @ A2 )
        = one_one_nat )
     => ~ ! [X5: nat] :
            ( A2
           != ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_8679_card__1__singletonE,axiom,
    ! [A2: set_int] :
      ( ( ( finite_card_int @ A2 )
        = one_one_nat )
     => ~ ! [X5: int] :
            ( A2
           != ( insert_int2 @ X5 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_8680_psubset__card__mono,axiom,
    ! [B2: set_set_nat,A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( ord_less_set_set_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_8681_psubset__card__mono,axiom,
    ! [B2: set_list_nat,A2: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B2 )
     => ( ( ord_le1190675801316882794st_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_8682_psubset__card__mono,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_set_nat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_8683_psubset__card__mono,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_less_set_complex @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_complex @ A2 ) @ ( finite_card_complex @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_8684_psubset__card__mono,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_set_int @ A2 @ B2 )
       => ( ord_less_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_8685_psubset__card__mono,axiom,
    ! [B2: set_Extended_enat,A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ B2 )
     => ( ( ord_le2529575680413868914d_enat @ A2 @ B2 )
       => ( ord_less_nat @ ( finite121521170596916366d_enat @ A2 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).

% psubset_card_mono
thf(fact_8686_card__distinct,axiom,
    ! [Xs: list_complex] :
      ( ( ( finite_card_complex @ ( set_complex2 @ Xs ) )
        = ( size_s3451745648224563538omplex @ Xs ) )
     => ( distinct_complex @ Xs ) ) ).

% card_distinct
thf(fact_8687_card__distinct,axiom,
    ! [Xs: list_set_nat] :
      ( ( ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) )
        = ( size_s3254054031482475050et_nat @ Xs ) )
     => ( distinct_set_nat @ Xs ) ) ).

% card_distinct
thf(fact_8688_card__distinct,axiom,
    ! [Xs: list_list_nat] :
      ( ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) )
        = ( size_s3023201423986296836st_nat @ Xs ) )
     => ( distinct_list_nat @ Xs ) ) ).

% card_distinct
thf(fact_8689_card__distinct,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) )
        = ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( distinct_VEBT_VEBT @ Xs ) ) ).

% card_distinct
thf(fact_8690_card__distinct,axiom,
    ! [Xs: list_int] :
      ( ( ( finite_card_int @ ( set_int2 @ Xs ) )
        = ( size_size_list_int @ Xs ) )
     => ( distinct_int @ Xs ) ) ).

% card_distinct
thf(fact_8691_card__distinct,axiom,
    ! [Xs: list_nat] :
      ( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
        = ( size_size_list_nat @ Xs ) )
     => ( distinct_nat @ Xs ) ) ).

% card_distinct
thf(fact_8692_distinct__card,axiom,
    ! [Xs: list_complex] :
      ( ( distinct_complex @ Xs )
     => ( ( finite_card_complex @ ( set_complex2 @ Xs ) )
        = ( size_s3451745648224563538omplex @ Xs ) ) ) ).

% distinct_card
thf(fact_8693_distinct__card,axiom,
    ! [Xs: list_set_nat] :
      ( ( distinct_set_nat @ Xs )
     => ( ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) )
        = ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% distinct_card
thf(fact_8694_distinct__card,axiom,
    ! [Xs: list_list_nat] :
      ( ( distinct_list_nat @ Xs )
     => ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) )
        = ( size_s3023201423986296836st_nat @ Xs ) ) ) ).

% distinct_card
thf(fact_8695_distinct__card,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs )
     => ( ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) )
        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% distinct_card
thf(fact_8696_distinct__card,axiom,
    ! [Xs: list_int] :
      ( ( distinct_int @ Xs )
     => ( ( finite_card_int @ ( set_int2 @ Xs ) )
        = ( size_size_list_int @ Xs ) ) ) ).

% distinct_card
thf(fact_8697_distinct__card,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
        = ( size_size_list_nat @ Xs ) ) ) ).

% distinct_card
thf(fact_8698_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat2 @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat2 @ ( suc @ K2 ) @ M7 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat2 @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_8699_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat2 @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat2 @ ( suc @ K2 ) @ M7 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat2 @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_8700_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat2 @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat2 @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_8701_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
     => ( A2
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_8702_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_8703_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_8704_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z6: complex] :
                  ( ( power_power_complex @ Z6 @ N2 )
                  = C ) ) )
          = N2 ) ) ) ).

% card_nth_roots
thf(fact_8705_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) ) )
        = N2 ) ) ).

% card_roots_unity_eq
thf(fact_8706_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
    @ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_8707_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_8708_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_8709_int_Oabs__induct,axiom,
    ! [P2: int > $o,X: int] :
      ( ! [Y3: product_prod_nat_nat] : ( P2 @ ( abs_Integ @ Y3 ) )
     => ( P2 @ X ) ) ).

% int.abs_induct
thf(fact_8710_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X5: nat,Y3: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_8711_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_8712_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_8713_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_8714_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_8715_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_8716_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_8717_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_8718_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_8719_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X2: nat,Y2: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_8720_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N6: nat] :
              ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S3 ) )
             => ( member_nat2 @ ( R3 @ N6 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_8721_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_8722_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y2: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X2 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_int.rep_eq
thf(fact_8723_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M2: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N ) ) @ M2 ) ) ) ).

% prod_encode_def
thf(fact_8724_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_8725_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_8726_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).

% nat.rep_eq
thf(fact_8727_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_8728_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) ) ) ) ).

% uminus_int_def
thf(fact_8729_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_8730_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_8731_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_8732_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_8733_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_8734_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_8735_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_8736_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_8737_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X2: nat,Y2: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_8738_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I5: nat] : ( minus_minus_nat @ I5 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I5: nat] : ( minus_minus_nat @ I5 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I5: nat] : ( minus_minus_nat @ I5 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_8739_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_8740_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_8741_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_8742_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_8743_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X2: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ) ).

% less_int_def
thf(fact_8744_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
        @ ^ [X2: nat,Y2: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_8745_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_8746_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_8747_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_8748_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_8749_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_8750_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_8751_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
      = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_8752_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_8753_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => ( member_nat2 @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).

% Inf_nat_def1
thf(fact_8754_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_8755_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_8756_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real2 @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_8757_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_8758_int__in__range__abs,axiom,
    ! [N2: nat] : ( member_int2 @ ( semiri1314217659103216013at_int @ N2 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).

% int_in_range_abs
thf(fact_8759_root__def,axiom,
    ( root
    = ( ^ [N: nat,X2: real] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
            @ X2 ) ) ) ) ).

% root_def
thf(fact_8760_DERIV__even__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_8761_DERIV__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_8762_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_8763_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_8764_DERIV__odd__real__root,axiom,
    ! [N2: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_8765_Maclaurin,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ 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 @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_8766_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ 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 @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_8767_Maclaurin__minus,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N2 )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ 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 @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_8768_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ 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 @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_8769_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,T6: real] :
            ( ( ( ord_less_nat @ M3 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_8770_Taylor__down,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A )
                    = ( 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 @ A @ C ) @ M2 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_8771_Taylor__up,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F @ B )
                    = ( 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 @ B @ C ) @ M2 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_8772_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N2 )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F @ X )
                          = ( 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 @ X @ C ) @ M2 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_8773_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
      ( ! [M3: nat,T6: real] :
          ( ( ( ord_less_nat @ M3 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M4: nat,T7: real] :
            ( ( ( ord_less_nat @ M4 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M4 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M4 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M4 ) ) )
                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M4 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M4 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M4 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M4 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M4 ) ) ) )
                  @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N2 @ ( suc @ M4 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M4 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_8774_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_8775_DERIV__real__root__generic,axiom,
    ! [N2: nat,X: real,D6: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D6
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D6
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
               => ( D6
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D6 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_8776_DERIV__pow,axiom,
    ! [N2: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X2: real] : ( power_power_real @ X2 @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_8777_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real2 @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X5 @ N ) ) ) )
     => ( ( member_real2 @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] :
                ( suminf_real
                @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) )
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_8778_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_8779_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_8780_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_8781_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N6: nat] :
              ( member_real2
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_8782_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N6: nat] :
              ( member_real2
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_8783_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_8784_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_8785_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_8786_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_8787_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_8788_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_8789_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
         => ( ( filterlim_nat_real
              @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N6: nat] : ( ord_less_eq_real @ L4 @ ( G @ N6 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_8790_LIMSEQ__inverse__zero,axiom,
    ! [X7: nat > real] :
      ( ! [R3: real] :
        ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_real @ R3 @ ( X7 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( X7 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_8791_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_8792_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_8793_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 )
             => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_8794_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_8795_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_8796_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).

% summable
thf(fact_8797_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_8798_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_8799_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_8800_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N6: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N6: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_8801_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_8802_eventually__sequentially__Suc,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat
        @ ^ [I5: nat] : ( P2 @ ( suc @ I5 ) )
        @ at_top_nat )
      = ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_8803_eventually__sequentially__seg,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N: nat] : ( P2 @ ( plus_plus_nat @ N @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_8804_eventually__sequentially,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat @ P2 @ at_top_nat )
      = ( ? [N5: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N5 @ N )
           => ( P2 @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_8805_eventually__sequentiallyI,axiom,
    ! [C: nat,P2: nat > $o] :
      ( ! [X5: nat] :
          ( ( ord_less_eq_nat @ C @ X5 )
         => ( P2 @ X5 ) )
     => ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_8806_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
      = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F3 ) ) ) ).

% le_sequentially
thf(fact_8807_sequentially__offset,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( eventually_nat @ P2 @ at_top_nat )
     => ( eventually_nat
        @ ^ [I5: nat] : ( P2 @ ( plus_plus_nat @ I5 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_8808_filterlim__pow__at__bot__even,axiom,
    ! [N2: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_8809_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_8810_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_8811_filterlim__pow__at__bot__odd,axiom,
    ! [N2: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_8812_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_8813_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_8814_mono__times__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).

% mono_times_nat
thf(fact_8815_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_8816_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_8817_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat2 @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_8818_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_8819_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_8820_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_8821_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat2 @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_8822_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat2 @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M2: nat] :
                      ( collect_nat
                      @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M2 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_8823_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_8824_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_8825_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_8826_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N ) @ M2 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_8827_range__abs__Nats,axiom,
    ( ( image_int_int @ abs_abs_int @ top_top_set_int )
    = semiring_1_Nats_int ) ).

% range_abs_Nats
thf(fact_8828_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_8829_inj__Suc,axiom,
    ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).

% inj_Suc
thf(fact_8830_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat2 @ N3 @ N7 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N7 ) ) ).

% inj_on_diff_nat
thf(fact_8831_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M2: nat,N: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ) ).

% pred_nat_def
thf(fact_8832_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                    & ! [X2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                        & ! [I5: nat] :
                            ( ( ord_less_nat @ I5 @ ( 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 @ TreeList3 @ I5 ) @ X8 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X2: nat] :
                                ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                 => ( ( ord_less_nat @ Mi3 @ X2 )
                                    & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_8833_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                          & ! [I5: nat] :
                              ( ( ord_less_nat @ I5 @ ( 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 @ TreeList3 @ I5 ) @ X8 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X2: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X2: nat] :
                                  ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                   => ( ( ord_less_nat @ Mi3 @ X2 )
                                      & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_8834_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X2 @ ( 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 )
            & ! [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                & ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( 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 @ I5 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I5 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X2: nat] :
                        ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_8835_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                              & ! [I5: nat] :
                                  ( ( ord_less_nat @ I5 @ ( 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 @ TreeList3 @ I5 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X2: nat] :
                                      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                       => ( ( ord_less_nat @ Mi3 @ X2 )
                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_8836_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                              & ! [I5: nat] :
                                  ( ( ord_less_nat @ I5 @ ( 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 @ TreeList3 @ I5 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X2: nat] :
                                      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                       => ( ( ord_less_nat @ Mi3 @ X2 )
                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_8837_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT2 @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                          & ! [X2: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                              & ! [I5: nat] :
                                  ( ( ord_less_nat @ I5 @ ( 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 @ TreeList3 @ I5 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X2: nat] :
                                      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                       => ( ( ord_less_nat @ Mi3 @ X2 )
                                          & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_8838_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT2 @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
                        & ! [X2: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ 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 ) )
                            & ! [I5: nat] :
                                ( ( ord_less_nat @ I5 @ ( 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 @ TreeList3 @ I5 ) @ X8 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT2 @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X2: nat] :
                                    ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
                                     => ( ( ord_less_nat @ Mi3 @ X2 )
                                        & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_8839_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_8840_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_8841_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_8842_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_8843_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M2: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_8844_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I5: int,N: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I5 ) @ ( semiri5074537144036343181t_real @ N ) ) )
          & ( N != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_8845_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_8846_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_8847_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_8848_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% min_numeral_Suc
thf(fact_8849_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_8850_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_8851_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_8852_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).

% min_diff
thf(fact_8853_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_8854_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_8855_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N2 @ M5 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_8856_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_8857_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_8858_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_8859_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
thf(fact_8860_binomial__def,axiom,
    ( binomial
    = ( ^ [N: nat,K2: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat2 @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K2 ) ) ) ) ) ) ).

% binomial_def
thf(fact_8861_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% less_eq
thf(fact_8862_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_8863_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_8864_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_8865_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_8866_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_8867_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs3: list_nat] :
              ( ( X
                = ( cons_nat @ X5 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_8868_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_8869_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_8870_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I5: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I5 ) @ Js @ ( upto_aux @ I5 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_8871_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_8872_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_8873_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_8874_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_8875_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_8876_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_8877_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_8878_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_8879_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_8880_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( 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 @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_8881_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( 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 @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_8882_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_8883_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_8884_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ I5 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_8885_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ I5 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_8886_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_8887_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I5: int,J3: int] : ( append_int @ ( upto @ I5 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_8888_upto__code,axiom,
    ( upto
    = ( ^ [I5: int,J3: int] : ( upto_aux @ I5 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_8889_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_8890_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_8891_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I5 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_8892_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa2 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa2 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_8893_upto_Osimps,axiom,
    ( upto
    = ( ^ [I5: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I5 @ J3 ) @ ( cons_int @ I5 @ ( upto @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_8894_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_8895_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_8896_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_8897_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X5: nat,Xs3: list_nat] :
                ( ( X
                  = ( cons_nat @ X5 @ Xs3 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_8898_remdups__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( remdups_nat @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% remdups_upt
thf(fact_8899_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_8900_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_8901_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_8902_take__upt,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
     => ( ( take_nat @ M @ ( upt @ I @ N2 ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_8903_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_8904_sorted__list__of__set__range,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sorted_list_of_set_range
thf(fact_8905_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_8906_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_8907_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_8908_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ I5 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_8909_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_8910_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_8911_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_8912_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M2 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_8913_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_8914_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_8915_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I5: nat,J3: nat] : ( set_nat2 @ ( upt @ I5 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_8916_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M2 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_8917_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M2 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_8918_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_8919_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_8920_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_8921_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_8922_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_8923_upt__rec,axiom,
    ( upt
    = ( ^ [I5: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I5 @ J3 ) @ ( cons_nat @ I5 @ ( upt @ ( suc @ I5 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_8924_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_8925_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_8926_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).

% sum_list_upt
thf(fact_8927_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: 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 )
                  = N7 ) ) ) )
        = ( 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 )
                    = N7 ) ) ) )
          @ ( 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 )
                    = N7 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_8928_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N7 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M ) @ one_one_nat ) @ N7 ) ) ).

% card_length_sum_list
thf(fact_8929_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_8930_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_8931_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_8932_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_8933_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_8934_sorted__upto,axiom,
    ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).

% sorted_upto
thf(fact_8935_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I5: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I5 @ J3 ) @ M ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
        @ ^ [R4: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R4 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_8936_vimage__Suc__insert__Suc,axiom,
    ! [N2: nat,A2: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat2 @ ( suc @ N2 ) @ A2 ) )
      = ( insert_nat2 @ N2 @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_8937_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
      = ( finite_finite_nat @ F3 ) ) ).

% finite_vimage_Suc_iff
thf(fact_8938_vimage__Suc__insert__0,axiom,
    ! [A2: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat2 @ zero_zero_nat @ A2 ) )
      = ( vimage_nat_nat @ suc @ A2 ) ) ).

% vimage_Suc_insert_0
thf(fact_8939_set__decode__div__2,axiom,
    ! [X: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).

% set_decode_div_2
thf(fact_8940_set__encode__vimage__Suc,axiom,
    ! [A2: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc
thf(fact_8941_GreatestI__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_nat
thf(fact_8942_Greatest__le__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% Greatest_le_nat
thf(fact_8943_GreatestI__ex__nat,axiom,
    ! [P2: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_8944_Field__natLeq__on,axiom,
    ! [N2: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y2: nat] :
                ( ( ord_less_nat @ X2 @ N2 )
                & ( ord_less_nat @ Y2 @ N2 )
                & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) )
      = ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N2 ) ) ) ).

% Field_natLeq_on
thf(fact_8945_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_8946_Restr__natLeq,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N2 ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N2 ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_8947_wf__int__ge__less__than2,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).

% wf_int_ge_less_than2
thf(fact_8948_wf__int__ge__less__than,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).

% wf_int_ge_less_than
thf(fact_8949_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_8950_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_8951_Restr__natLeq2,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
          @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_8952_natLeq__underS__less,axiom,
    ! [N2: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
      = ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N2 ) ) ) ).

% natLeq_underS_less
thf(fact_8953_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_8954_bij__betw__Suc,axiom,
    ! [M7: set_nat,N7: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N7 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N7 ) ) ).

% bij_betw_Suc
thf(fact_8955_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_8956_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X2: nat] : X2
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_8957_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X2: int] : X2
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_8958_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).

% strict_mono_imp_increasing
thf(fact_8959_le__enumerate,axiom,
    ! [S3: set_nat,N2: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ).

% le_enumerate
thf(fact_8960_finite__le__enumerate,axiom,
    ! [S3: set_nat,N2: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ N2 @ ( finite_card_nat @ S3 ) )
       => ( ord_less_eq_nat @ N2 @ ( infini8530281810654367211te_nat @ S3 @ N2 ) ) ) ) ).

% finite_le_enumerate
thf(fact_8961_Least__eq__0,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ zero_zero_nat )
     => ( ( ord_Least_nat @ P2 )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_8962_Least__Suc,axiom,
    ! [P2: nat > $o,N2: nat] :
      ( ( P2 @ N2 )
     => ( ~ ( P2 @ zero_zero_nat )
       => ( ( ord_Least_nat @ P2 )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M2: nat] : ( P2 @ ( suc @ M2 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_8963_Least__Suc2,axiom,
    ! [P2: nat > $o,N2: nat,Q: nat > $o,M: nat] :
      ( ( P2 @ N2 )
     => ( ( Q @ M )
       => ( ~ ( P2 @ zero_zero_nat )
         => ( ! [K3: nat] :
                ( ( P2 @ ( suc @ K3 ) )
                = ( Q @ K3 ) )
           => ( ( ord_Least_nat @ P2 )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_8964_eventually__prod__sequentially,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P2 @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N5: nat] :
          ! [M2: nat] :
            ( ( ord_less_eq_nat @ N5 @ M2 )
           => ! [N: nat] :
                ( ( ord_less_eq_nat @ N5 @ N )
               => ( P2 @ ( product_Pair_nat_nat @ N @ M2 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_8965_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_8966_natLeq__on__wo__rel,axiom,
    ! [N2: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_8967_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N: nat] :
        ( ( dvd_dvd_nat @ M2 @ N )
        & ( M2 != N ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_8968_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
    @ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_8969_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_8970_less__enat__def,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [M2: extended_enat,N: extended_enat] :
          ( extended_case_enat_o
          @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N )
          @ $false
          @ M2 ) ) ) ).

% less_enat_def
thf(fact_8971_natLeq__on__well__order__on,axiom,
    ! [N2: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N2 ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_8972_natLeq__on__Well__order,axiom,
    ! [N2: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y2: nat] :
                ( ( ord_less_nat @ X2 @ N2 )
                & ( ord_less_nat @ Y2 @ N2 )
                & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N2 )
              & ( ord_less_nat @ Y2 @ N2 )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_8973_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) )
      = ( nat_prod_decode_aux @ K @ M ) ) ).

% prod_decode_triangle_add
thf(fact_8974_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_8975_list__decode_Opinduct,axiom,
    ! [A0: nat,P2: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
       => ( ! [N3: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
             => ( ! [X3: nat,Y4: nat] :
                    ( ( ( product_Pair_nat_nat @ X3 @ Y4 )
                      = ( nat_prod_decode @ N3 ) )
                   => ( P2 @ Y4 ) )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_8976_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N3: nat] :
              ( ( X
                = ( suc @ N3 ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X2: nat,Y2: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y2 ) )
                  @ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_8977_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_8978_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_8979_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
     => ( ( nat_list_decode @ ( suc @ N2 ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X2: nat,Y2: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y2 ) )
          @ ( nat_prod_decode @ N2 ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_8980_list__decode_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( nat_list_decode @ ( suc @ N2 ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X2: nat,Y2: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y2 ) )
        @ ( nat_prod_decode @ N2 ) ) ) ).

% list_decode.simps(2)
thf(fact_8981_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N3: nat] :
                ( ( X
                  = ( suc @ N3 ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X2: nat,Y2: nat] : ( cons_nat @ X2 @ ( nat_list_decode @ Y2 ) )
                      @ ( nat_prod_decode @ N3 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_8982_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_8983_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_8984_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_8985_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_8986_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_8987_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_8988_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( ( member_real2 @ X @ field_5140801741446780682s_real )
     => ~ ! [M3: nat,N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M3 @ N3 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_8989_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_8990_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_8991_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_8992_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_8993_coprime__Suc__0__left,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N2 ) ).

% coprime_Suc_0_left
thf(fact_8994_coprime__Suc__0__right,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_8995_coprime__Suc__left__nat,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ ( suc @ N2 ) @ N2 ) ).

% coprime_Suc_left_nat
thf(fact_8996_coprime__Suc__right__nat,axiom,
    ! [N2: nat] : ( algebr934650988132801477me_nat @ N2 @ ( suc @ N2 ) ) ).

% coprime_Suc_right_nat
thf(fact_8997_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_8998_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_8999_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ N2 ) ) ).

% coprime_diff_one_left_nat
thf(fact_9000_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_9001_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_9002_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ pcr_int
    @ ^ [N: nat] : ( product_Pair_nat_nat @ N @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9003_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_9004_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X @ V )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_9005_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y5: nat,Z3: nat] : Y5 = Z3
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_9006_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 ) ) ) ).

% uminus_int.rsp
thf(fact_9007_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_9008_int_Oabs__eq__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( abs_Integ @ X )
        = ( abs_Integ @ Y ) )
      = ( intrel @ X @ Y ) ) ).

% int.abs_eq_iff
thf(fact_9009_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_9010_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] :
              ( ( plus_plus_nat @ X2 @ V4 )
              = ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% intrel_def
thf(fact_9011_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% less_int.rsp
thf(fact_9012_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X2: nat,Y2: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_9013_int_Orel__eq__transfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z3: $o] : Y5 = Z3 )
    @ intrel
    @ ^ [Y5: int,Z3: int] : Y5 = Z3 ) ).

% int.rel_eq_transfer
thf(fact_9014_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_9015_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X2: nat,Y2: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_9016_int_Obi__total,axiom,
    bi_tot896582865486249351at_int @ pcr_int ).

% int.bi_total
thf(fact_9017_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_9018_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimnum
thf(fact_9019_idiff__enat__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ ( extended_enat2 @ zero_zero_nat ) )
      = N2 ) ).

% idiff_enat_0_right
thf(fact_9020_idiff__enat__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N2 )
      = ( extended_enat2 @ zero_zero_nat ) ) ).

% idiff_enat_0
thf(fact_9021_plus__enat__simps_I1_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% plus_enat_simps(1)
thf(fact_9022_max__enat__simps_I1_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_ma741700101516333627d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_enat_simps(1)
thf(fact_9023_enat__ord__simps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% enat_ord_simps(2)
thf(fact_9024_enat__ord__simps_I1_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% enat_ord_simps(1)
thf(fact_9025_numeral__less__enat__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% numeral_less_enat_iff
thf(fact_9026_numeral__le__enat__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% numeral_le_enat_iff
thf(fact_9027_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_9028_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = zero_z5237406670263579293d_enat )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_9029_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X ) )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_9030_less__enatE,axiom,
    ! [N2: extended_enat,M: nat] :
      ( ( ord_le72135733267957522d_enat @ N2 @ ( extended_enat2 @ M ) )
     => ~ ! [K3: nat] :
            ( ( N2
              = ( extended_enat2 @ K3 ) )
           => ~ ( ord_less_nat @ K3 @ M ) ) ) ).

% less_enatE
thf(fact_9031_Suc__ile__eq,axiom,
    ! [M: nat,N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N2 )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N2 ) ) ).

% Suc_ile_eq
thf(fact_9032_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N2: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N2 ) )
      = ( ? [Y6: nat,X9: nat] :
            ( ( X
              = ( extended_enat2 @ X9 ) )
            & ( Y
              = ( extended_enat2 @ Y6 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y6 ) @ N2 ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_9033_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L ) )
      = ( vEBT_Node @ Info @ Deg
        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( map_VE8901447254227204932T_VEBT
            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
            @ TreeList2 ) )
        @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.simps(3)
thf(fact_9034_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B4: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B4 ) )
           => ( Y
             != ( vEBT_Leaf @ A5 @ B4 ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y
                 != ( vEBT_Node @ Info2 @ Deg2
                    @ ( map_VE8901447254227204932T_VEBT
                      @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      @ TreeList3 )
                    @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ! [L4: nat] :
                    ( ( Xa2
                      = ( extended_enat2 @ L4 ) )
                   => ( Y
                     != ( vEBT_Node @ Info2 @ Deg2
                        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ( map_VE8901447254227204932T_VEBT
                            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            @ TreeList3 ) )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
thf(fact_9035_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimcomplete
thf(fact_9036_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_9037_times__enat__simps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N2 ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N2 ) )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
thf(fact_9038_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A5: $o,B4: $o,Uu2: extended_enat] :
          ( X
         != ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B4 ) @ Uu2 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ extend5688581933313929465d_enat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,L4: nat] :
              ( X
             != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).

% VEBT_internal.elim_dead.cases
thf(fact_9039_Sup__enat__def,axiom,
    ( comple4398354569131411667d_enat
    = ( ^ [A4: set_Extended_enat] : ( if_Extended_enat @ ( A4 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A4 ) @ ( lattic921264341876707157d_enat @ A4 ) @ extend5688581933313929465d_enat ) ) ) ) ).

% Sup_enat_def
thf(fact_9040_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_9041_Inf__enat__def,axiom,
    ( comple2295165028678016749d_enat
    = ( ^ [A4: set_Extended_enat] :
          ( if_Extended_enat @ ( A4 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
          @ ( ord_Le1955565732374568822d_enat
            @ ^ [X2: extended_enat] : ( member_Extended_enat @ X2 @ A4 ) ) ) ) ) ).

% Inf_enat_def
thf(fact_9042_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
      = ( vEBT_Node @ Info @ Deg
        @ ( map_VE8901447254227204932T_VEBT
          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          @ TreeList2 )
        @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).

% VEBT_internal.elim_dead.simps(2)
thf(fact_9043_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X @ Xa2 ) )
       => ( ! [A5: $o,B4: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B4 ) )
             => ( ( Y
                  = ( vEBT_Leaf @ A5 @ B4 ) )
               => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A5 @ B4 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ Deg2
                        @ ( map_VE8901447254227204932T_VEBT
                          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ TreeList3 )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
                   => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ! [L4: nat] :
                      ( ( Xa2
                        = ( extended_enat2 @ L4 ) )
                     => ( ( Y
                          = ( vEBT_Node @ Info2 @ Deg2
                            @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                              @ ( map_VE8901447254227204932T_VEBT
                                @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ TreeList3 ) )
                            @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
thf(fact_9044_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M2: extended_enat,N: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P6: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P6 ) )
              @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
              @ N )
          @ ( if_Extended_enat @ ( N = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M2 ) ) ) ).

% times_enat_def
thf(fact_9045_plus__enat__def,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [M2: extended_enat,N: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P6: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P6 ) )
              @ extend5688581933313929465d_enat
              @ N )
          @ extend5688581933313929465d_enat
          @ M2 ) ) ) ).

% plus_enat_def
thf(fact_9046_eSuc__Max,axiom,
    ! [A2: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( A2 != bot_bo7653980558646680370d_enat )
       => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A2 ) )
          = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A2 ) ) ) ) ) ).

% eSuc_Max
thf(fact_9047_eSuc__def,axiom,
    ( extended_eSuc
    = ( extend3600170679010898289d_enat
      @ ^ [N: nat] : ( extended_enat2 @ ( suc @ N ) )
      @ extend5688581933313929465d_enat ) ) ).

% eSuc_def
thf(fact_9048_eSuc__enat,axiom,
    ! [N2: nat] :
      ( ( extended_eSuc @ ( extended_enat2 @ N2 ) )
      = ( extended_enat2 @ ( suc @ N2 ) ) ) ).

% eSuc_enat
thf(fact_9049_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( ( extended_eSuc @ X )
        = ( extended_enat2 @ Y ) )
      = ( ? [N: nat] :
            ( ( Y
              = ( suc @ N ) )
            & ( X
              = ( extended_enat2 @ N ) ) ) ) ) ).

% eSuc_enat_iff
thf(fact_9050_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( ( extended_enat2 @ Y )
        = ( extended_eSuc @ X ) )
      = ( ? [N: nat] :
            ( ( Y
              = ( suc @ N ) )
            & ( ( extended_enat2 @ N )
              = X ) ) ) ) ).

% enat_eSuc_iff
thf(fact_9051_eSuc__Sup,axiom,
    ! [A2: set_Extended_enat] :
      ( ( A2 != bot_bo7653980558646680370d_enat )
     => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A2 ) )
        = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A2 ) ) ) ) ).

% eSuc_Sup
thf(fact_9052_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o2 @ $false @ ( insert_o2 @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_9053_Rep__unit__induct,axiom,
    ! [Y: $o,P2: $o > $o] :
      ( ( member_o2 @ Y @ ( insert_o2 @ $true @ bot_bot_set_o ) )
     => ( ! [X5: product_unit] : ( P2 @ ( product_Rep_unit @ X5 ) )
       => ( P2 @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_9054_Rep__unit,axiom,
    ! [X: product_unit] : ( member_o2 @ ( product_Rep_unit @ X ) @ ( insert_o2 @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_9055_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( ( member_o2 @ Y @ ( insert_o2 @ $true @ bot_bot_set_o ) )
     => ~ ! [X5: product_unit] :
            ( Y
            = ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).

% Rep_unit_cases
thf(fact_9056_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( ( member_o2 @ Y @ ( insert_o2 @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
        = Y ) ) ).

% Abs_unit_inverse
thf(fact_9057_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( ( member_o2 @ X @ ( insert_o2 @ $true @ bot_bot_set_o ) )
     => ( ( member_o2 @ Y @ ( insert_o2 @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X )
            = ( product_Abs_unit @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_unit_inject
thf(fact_9058_Abs__unit__induct,axiom,
    ! [P2: product_unit > $o,X: product_unit] :
      ( ! [Y3: $o] :
          ( ( member_o2 @ Y3 @ ( insert_o2 @ $true @ bot_bot_set_o ) )
         => ( P2 @ ( product_Abs_unit @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% Abs_unit_induct
thf(fact_9059_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y3: $o] :
          ( ( X
            = ( product_Abs_unit @ Y3 ) )
         => ~ ( member_o2 @ Y3 @ ( insert_o2 @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_9060_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o2 @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_9061_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9062_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9063_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9064_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9065_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9066_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9067_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9068_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9069_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( member_nat2 @ A5 @ A2 )
         => ( ( member_nat2 @ B4 @ A2 )
           => ( member_nat2 @ ( gcd_gcd_nat @ A5 @ B4 ) @ A2 ) ) )
     => ( ( A2 != bot_bot_set_nat )
       => ( member_nat2 @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).

% Gcd_in
thf(fact_9070_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_9071_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_9072_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9073_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_9074_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9075_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X2: nat,Y2: nat] : ( if_nat @ ( Y2 = zero_zero_nat ) @ X2 @ ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9076_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9077_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9078_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X5: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9079_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_9080_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N: nat] :
        ( ( dvd_dvd_nat @ M2 @ N )
        & ( M2 != N ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9081_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D4: nat] :
                ( ( dvd_dvd_nat @ D4 @ M )
                & ( dvd_dvd_nat @ D4 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9082_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9083_cr__int__def,axiom,
    ( cr_int
    = ( ^ [X2: product_prod_nat_nat] :
          ( ^ [Y5: int,Z3: int] : Y5 = Z3
          @ ( abs_Integ @ X2 ) ) ) ) ).

% cr_int_def
thf(fact_9084_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
thf(fact_9085_Quotient__int,axiom,
    quotie1194848508323700631at_int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).

% Quotient_int

% Helper facts (31)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( if_filter_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( if_filter_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ? [Y4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ i ) @ Y4 ) ).

%------------------------------------------------------------------------------